Past News and Events

BMO Round 1 marked (11–14 December 2009)

BMO Round 1 was marked in Cambridge on 11–13 December; 17 candidates in New Zealand also sat the paper, their scripts were marked in New Zealand using the same mark schemes as used in the UK, and such candidates are marked “(NZ)” below. Results and marked scripts will be sent to schools by mid-January. The following candidates scored 44 or more out of 60.

BMO Round 1 (4 December 2009)

This year’s BMO Round 1 paper was taken yesterday; video solutions are temporarily available online.

BMO entry fees (29 September 2009)

With immediate effect, the BMO1 entry fee will be £16.50 and the BMO2 fee will be £22.00. This is a 10% increase, the first for three years, and is exactly in line with the price increases for UKMT Maths Challenges competitions. Entry fees are waived for candidates eligible to represent the UK at the IMO and reaching the cut-off scores in qualifying competitions (UK SMC for entry to BMO1, BMO1 for entry to BMO2). Schools should contact the Maths Challenges Office to enter candidates who do not qualify for waiver of fees.

Royal Society celebration (21 September 2009)

The 2009 olympiad teams’ achievements were celebrated at the Royal Society on Monday 21 September; Professor James Davenport gave the 32nd annual IMO lecture, on “The Mathematics behind the Internet” (slides available).

Balkan Mathematical Olympiad report available (1 August 2009)

The Balkan Mathematical Olympiad information now includes Vesna Kadelburg’s report on the 2009 olympiad.

The involvement of the UK team in this competition is sponsored by Winton Capital Management.

IMO 2009 report available (28 July 2009)

The reports on past IMOs now include Geoff Smith’s report on IMO 2009.

IMO 2009 results (19 July 2009)

The IMO papers were sat on 15 and 16 July. The results of the UK team members are shown below. We came equal 19th out of 104 countries competing with 1 Gold, 3 Silver and 2 Bronze medals and a score of 157 out of 252. The medal boundaries were for 32 for Gold, 24 for Silver and 14 for Bronze.

IMO 2009 problems (16 July 2009)

The IMO papers were set on 15 and 16 July; the problems are available.

Mathematical Ashes results (9 July 2009)

The UK has won the second Mathematical Ashes competition, decided on the results of today’s practice exam at the joint pre-IMO training camp in Cambridge; the results for the two teams were as follows:

IMO team selected (27 May 2009)

After the training camp held at Oundle School from 23–27 May, the team of six for the IMO (to be held in Bremen, Germany from 10–22 July) has been chosen as:

The three reserves are:

The Team Leader is Dr Geoff Smith of the University of Bath and the Deputy Leader is Dr Vesna Kadelburg of Mander Portman Woodward, Cambridge. James Cranch of the University of Leicester will be Observer with Leader, Mary Wimbury of the UKMT will be Observer with Deputy and Jacqui Lewis of St. Julian’s International School, Carcavelos, Portugal will be Observer with Contestants.

Balkan Mathematical Olympiad results (3 May 2009)

The UK team in the Balkan Mathematical Olympiad in Kragujevac, Serbia have achieved the following scores (each question marked out of 10). The team total of 59 placed us equal 13th out of 20 participating (official and guest) teams. The medal cut-offs were 23 for gold, 15 for silver and 6 for bronze.

The involvement of the UK team in this competition is sponsored by Winton Capital Management.

Balkan Mathematical Olympiad problems (30 April 2009)

The problems from this year’s Balkan Mathematical Olympiad, sat today, are available from the Mathematical Society of Serbia website.

The involvement of the UK team in this competition is sponsored by Winton Capital Management.

Balkan Mathematical Olympiad team selected (6 April 2009)

The UK has been invited as a guest nation to send a team to the Balkan Mathematical Olympiad, to be held in Kragujevac, Serbia from 28 April–4 May. The Team Leader is Dr Vesna Kadelburg of Mander Portman Woodward, Cambridge and the Deputy Leader is Robin Bhattacharyya of Loughborough Grammar School. After the training camp held at Trinity College, Cambridge from 2–6 April, the team has been selected as follows.

The involvement of the UK team in this competition is sponsored by Winton Capital Management.

IMO squad selected (6 April 2009)

After the training camp held at Trinity College, Cambridge from 2–6 April, the squad of nine from which the team of six and three reserves for the IMO in Bremen, Germany will be selected has been chosen as:

The Team Leader is Dr Geoff Smith of the University of Bath and the Deputy Leader is Dr Vesna Kadelburg of Mander Portman Woodward, Cambridge. The IMO will take place from 10–22 July.

Romanian Master in Mathematics 2009 report available (14 March 2009)

The Romanian Master in Mathematics information now includes Adrian Sanders’s report on RMM 2009.

Romanian Master in Mathematics results (1 March 2009)

The UK team in the Romanian Master in Mathematics competition have achieved the following results. In the team competition (determined based on the top three scores from each country), China won, with the USA and Serbia tied for second place and the UK tenth.

The involvement of the UK team in this competition is sponsored by Winton Capital Management.

BMO Round 2 sat by New Zealand squad (25 February 2009)

BMO Round 2 was sat by the members of the squad in training for the 2009 New Zealand IMO team, in addition to candidates in the UK. The results of the top New Zealand candidates are available from the New Zealand Maths Olympiad Committee site.

BMO Round 2 marked (8 February 2009)

BMO Round 2 was marked in Bath on 7 February; invitations to the Trinity training camp will be sent shortly to selected candidates eligible to represent the UK at the IMO, and details of marks will be sent to schools. The following candidates scored 25 or more out of 40.

Romanian Masters of Mathematics 2009 team (8 February 2009)

The UK will compete again in the Romanian Masters of Mathematics competition in 2009, to be held from 26 February to 2 March. The Team Leader will be Dr Adrian Sanders (formerly of Trinity College, Cambridge) and the Deputy Leader will be Mrs Sally Anne Huk (formerly of Haberdashers’ Aske’s School for Girls). This year teams of up to six students from each country sit one five-hour paper with four IMO-standard questions, with the top three scores from each country determining the result of the team competition. The UK team of six will be:

The United Kingdom are reigning champions. The involvement of the UK team in this competition is sponsored by Winton Capital Management.

BMO Round 2 (29 January 2009)

This year’s BMO Round 2 paper was taken today.

BMO Round 2 discussion embargo (28 January 2009)

There are candidates sitting BMO2 outside the United Kingdom this year. They will commence sitting the paper at 8:30pm UK time on Thursday 29 January. Therefore we ask that no public discussion of any aspect of the contents of BMO2 take place until after that time. The paper will appear at this site after that time.

Geoff Smith
for the BMO Committee.

BMO Round 2 typo (20 January 2009)

BMO Round 2 will be sat on 29 January. There is a mistake in the version of the paper, printed on white paper, originally distributed to schools. UKMT has posted corrected versions of the paper to participating schools today, so the corrected versions will arrive in schools very soon.

The corrected version is on blue paper. Candidates should raise a query if they are given a BMO2 paper to sit which is on white paper.

BMO Round 1 marked (12–14 December 2008)

BMO Round 1 was marked in Cambridge on 12–14 December. Results and marked scripts will be sent to schools in the week starting 5 January. The following candidates scored 41 or more out of 60.

BMO Round 1 (5 December 2008)

This year’s BMO Round 1 paper was taken yesterday, and today where examination centres were closed yesterday. See also information on post-BMO1 Masterclasses.

BMO Round 1 discussion embargo (4 December 2008)

We know that several BMO examination centres in northern England are closed for the day. There are probably other closures that we do not know about yet. (See the notice regarding arrangements for late sitting of BMO1 below.)

Accordingly, please do not discuss any aspect of BMO1 2008/9 in a public forum until 8:30pm UK time on Friday 5 December. The paper will appear at this site after that time.

The only exception is Dr Gardiner’s Birmingham masterclass which will go ahead as scheduled late on Friday afternoon. See information on post-BMO1 Masterclasses.

Geoff Smith
for the BMO Committee

Severe weather warnings and BMO Round 1 (3 December 2008)

There are currently severe weather warnings for some areas for Thursday 4 December. It is possible that there may be school closures, and students in affected areas may be unable to attend school on Thursday.

On no account should a student or teacher place him or herself in danger merely in order to facilitate the sitting of BMO1. Here is the BMO committee’s advice in the event that weather interferes with the sitting of BMO1 tomorrow (Thursday 4 December):

In the event of weather preventing a candidate from sitting BMO1 on 4 December, he or she should

(a) Refrain from using the internet until after the paper is sat.

(b) Do not discuss the paper with anyone.

(c) Sit the paper as soon as possible after 4 December 2008.

Would schoolteachers please include explanatory notes regarding late scripts.

The BMO committee reserves the right to withhold prizes if they suspect that the integrity of the competition may have been compromised.

Geoff Smith
for the BMO Committee

Romanian Master of Mathematics 2009 (11 November 2008)

The UK will compete again in the Romanian Master of Mathematics competition in 2009, to be held from 26 February to 2 March. The team will be accompanied by Adrian Sanders (ex Trinity College, Cambridge) and Sally Anne Huk (ex Haberdashers’ Aske’s School for Girls).

IMO talks on Australian radio (11 November 2008)

Some talks on the IMO that UK IMO Team Leader Geoff Smith recorded for Australian ABC Radio National are now being broadcast. The first talk was broadcast on Saturday 8 November and you can listen to it online. The second and third talks will be broadcast on 15 and 22 November.

Royal Society celebration (7 October 2008)

The IMO and Romanian Master in Mathematics teams’ achievements were celebrated at the Royal Society on Tuesday 7 October; Professor David Spiegelhalter gave the 31st annual IMO lecture, on “How long do we have to wait?”.

IMO 2008 report available (29 July 2008)

The reports on past IMOs now include Geoff Smith’s report on IMO 2008.

IMO 2008 results (20 July 2008)

The IMO papers were sat on 16 and 17 July. The results of the UK team members are shown below. We came 23rd out of 97 countries competing with four Silver and two Bronze medals and a score of 133 out of 252. The medal boundaries were for 31 for Gold, 22 for Silver and 15 for Bronze.

IMO papers (17 July 2008)

The IMO papers were sat on 16 and 17 July, and are available from the IMO 2008 website.

Mathematical Ashes results (14 July 2008)

Australia has won the inaugural Mathematical Ashes competition, decided on the results of the final practice exam at the joint pre-IMO training camp in Lisbon; the results for the two teams were as follows:

For more details of the competition and a picture of the urn in which the ashes will be placed, see the news story from the Australian Mathematics Trust.

IMO team selected (28 May 2008)

After the training camp held at Oundle School from 24–28 May 2008, the team of six for the IMO (to be held in Madrid, Spain from 10–22 July) has been chosen as:

The three reserves are:

The Team Leader is Dr Geoff Smith of the University of Bath, the Deputy Leader is Dr Ceri Fiddes of Stowe School, the Observer with Deputy is Dr Vesna Kadelburg of Sevenoaks School and the Observer with Contestants is Jacqui Lewis of St. Julian’s International School, Carcavelos, Portugal.

Balkan Mathematical Olympiad photographs (16 May 2008)

Some photographs from the United Kingdom and Ireland team and leadership at the Balkan Mathematical Olympiad are now available online. More are likely to be added in the coming days.

Balkan Mathematical Olympiad results (8 May 2008)

The United Kingdom and Ireland team in the Balkan Mathematical Olympiad in Ohrid have achieved the following scores (each question marked out of 10). The team total of 73 placed us 8th out of 19 participating (official and guest) teams.

The medal cut-offs were 29 for gold, 17 for silver and 5 for bronze. The problems are available on Mathlinks.

Balkan Mathematical Olympiad problems (6 May 2008)

The problems from today’s Balkan Mathematical Olympiad paper are available from Mathlinks.

Balkan Mathematical Olympiad team selected (11 April 2008)

A joint “United Kingdom and Ireland” team will be competing as a guest team in the Balkan Mathematical Olympiad, to be held in Ohrid, Macedonia from 4–10 May. The Team Leader is Dr Adrian Sanders (ex Trinity College, Cambridge) and the Deputy Leader is Jacqui Lewis (St. Julian’s International School, Carcavelos, Portugal). After the training camp held at Trinity College, Cambridge from 3–7 April, the six team members have been selected as follows:

Article on areal co-ordinates (9 April 2008)

A brief article on areal co-ordinate methods in Euclidean geometry, by Tom Lovering, is now available.

IMO squad selected (7 April 2008)

After the training camp held at Trinity College, Cambridge from 3–7 April, the squad of nine from which the team of six and three reserves for the IMO in Madrid, Spain will be selected has been chosen as:

The Team Leader is Dr Geoff Smith of the University of Bath, the Deputy Leader is Dr Ceri Fiddes of Stowe School, the Observer with Deputy is Dr Vesna Kadelburg of Sevenoaks School and the Observer with Contestants is Jacqui Lewis of St. Julian’s International School, Carcavelos, Portugal. The IMO will take place from 10–22 July.

Balkan Mathematical Olympiad team mostly selected (7 April 2008)

The UK has been invited as a guest nation to send a team to the Balkan Mathematical Olympiad, to be held in Ohrid, Macedonia from 4–10 May. The Team Leader is Dr Adrian Sanders (ex Trinity College, Cambridge) and the Deputy Leader is Jacqui Lewis (St. Julian’s International School, Carcavelos, Portugal). After the training camp held at Trinity College, Cambridge from 3–7 April, five of the six team members have been selected as follows; questions of availability for a sixth candidate team member are being resolved.

BMO Round 2 hints and outline solutions (21 February 2008)

Some hints and outline solutions for BMO Round 2 are now temporarily available. These are copyright and must not be redistributed.

Romanian Master in Mathematics report available (21 February 2008)

The Romanian Master of Mathematics information now includes Geoff Smith’s report on this competition.

BMO Round 2 marked (16 February 2008)

BMO Round 2 was marked in Bath on 16 February; invitations to the Trinity training camp will be sent shortly to selected candidates eligible to represent the UK at the IMO, and details of marks will be sent to schools. The following candidates scored 20 or more out of 40.

Romanian Master in Mathematics press materials (14 February 2008)

A press release and photos of the Romanian Master in Mathematics team for press use are now available.

Romanian Master in Mathematics results (10–11 February 2008)

The UK team in the Romanian Master in Mathematics competition have achieved the following results. We came first out of twelve teams competing with one Gold, one Silver and one Bronze medal and a total score of 51 out of 84. Jonathan Lee’s individual score of 24 was third out of the participating students.

Romanian Master in Mathematics paper (9 February 2008)

The Romanian Master in Mathematics 2008 paper is being sat today. Problem 1 is a UK submission and the other three problems are Romanian.

BMO Round 2 (31 January 2008)

This year’s BMO Round 2 paper was taken today.

“Romanian Master in Mathematics” team (30 January 2008)

The UK has been invited to send a team to the “Romanian Master in Mathematics” competition, to be held in Bucharest from 7–11 February. This competition has teams of three students per country sitting one five-hour paper with four IMO-standard questions. The Team Leader is Dr Geoff Smith of the University of Bath, and the Observer is Jacqui Lewis of St. Julian’s International School, Carcavelos, Portugal. After one originally selected student had to withdraw, the team of three will now be:

“Romanian Master in Mathematics” team selected (4–10 January 2008)

The UK has been invited to send a team to the “Romanian Master in Mathematics” competition, to be held in Bucharest from 7–11 February. This competition has teams of three students per country sitting one five-hour paper with four IMO-standard questions. The Team Leader is Dr Geoff Smith of the University of Bath, and the Observer is Jacqui Lewis of St. Julian’s International School, Carcavelos, Portugal. After the joint training camp held with the Hungarian IMO squad in Hungary over the New Year, the team of three has been selected as:

BMO Round 1 results distribution and BMO Round 2 cut-offs (18 December 2007)

BMO1 results will be sent to schools early in the new year. The BMO2 qualifying cut-off scores are 33 for students in year 13, 27 for students in year 12, 24 for students in year 11 and 20 for students in year 10 and below. Scoring at least the cut-off on BMO1 gives free entry to BMO2 provided that the candidate is eligible to represent the UK at the IMO. Any participant in BMO1 may enter BMO2 by paying the entry fee of £20. BMO2 will take place on 31 January 2008.

BMO Round 1 marked (7–9 December 2007)

BMO Round 1 was marked in Cambridge on 7–9 December. Details of cut-off scores for free entry to BMO Round 2 and when scores are to be sent to schools will appear here soon. The following candidates scored 48 or more out of 60.

BMO Round 1 (30 November 2007)

This year’s BMO Round 1 paper was taken today.

“Beautiful Young Minds” (14 October 2007)

The 90-minute documentary “Beautiful Young Minds”, filmed by Blast! Films during the team selection and training for IMO 2006 and at the IMO in Slovenia, was broadcast on BBC2 on Sunday 14 October at 9pm. See more details on Plus.

Royal Society celebration (1 October 2007)

The IMO and Balkan MO teams’ achievements were celebrated at the Royal Society on Monday 1 October; Professor Marcus du Sautoy gave the 30th annual IMO lecture, on “The Music of the Primes”.

IMO 2007 report available (12 August 2007)

The reports on past IMOs now include Geoff Smith’s report on IMO 2007.

IMO 2007 results (28 July 2007)

The IMO papers were sat on 25 and 26 July. The results of the UK team members are shown below. We came 28th out of 93 countries competing with one Gold and three Bronze medals and two Honourable Mentions and a score of 95 out of 252. The medal boundaries were for 29 for Gold, 21 for Silver and 14 for Bronze.

IMO Day 2 (26 July 2007)

The second IMO 2007 paper was sat in Hanoi on 26 July 2007; the problems and the countries submitting them are shown. Coordination takes place on 27–28 July and the final Jury meeting to approve scores and determine medal boundaries is scheduled for 22:00 local time (16:00 BST) on 28 July.

4. In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the perpendicular bisector of BC at P, and the perpendicular bisector of AC at Q. The midpoint of BC is K and the midpoint of AC is L. Prove that the triangles RPK and RQL have the same area.

   (Czech Republic)

5. Let a and b be positive integers. Show that if 4ab − 1 divides (4a² − 1)², then a = b.

   (United Kingdom)

6. Let n be a positive integer. Consider

   S = { (x, y, z) : x, y, z ∈ {0, 1, …, n}, x + y + z > 0 }

   as a set of (n + 1)³ − 1 points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains S but does not include (0, 0, 0).

   (Netherlands)

IMO Day 1 (25 July 2007)

The first IMO 2007 paper was sat in Hanoi on 25 July 2007; the problems and the countries submitting them are shown.

1. Real numbers a₁, a₂, …, a_n are given. For each i (1 ≤ i ≤ n) define

   d_i = max { a_j : 1 ≤ j ≤ i } − min { a_j : i ≤ j ≤ n }

   and let

   d = max { d_i : 1 ≤ i ≤ n }.

   (a) Prove that, for any real numbers x₁ ≤ x₂ ≤ … ≤ x_n,

   max { |x_i − a_i| : 1 ≤ i ≤ n } ≥ d / 2.    (*)

   (b) Show that there are real numbers x₁ ≤ x₂ ≤ … ≤ x_n such that equality holds in (*).

   (New Zealand)

2. Consider five points A, B, C, D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let ℓ be a line passing through A. Suppose that ℓ intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that EF = EG = EC. Prove that ℓ is the bisector of angle DAB.

   (Luxembourg)

3. In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size.

   Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

   (Russia)

IMO team selected (30 May 2007)

After the training camp held at Oundle School from 26–30 May 2007, the team of six for the IMO (to be held in Hanoi, Vietnam from 19–31 July) has been chosen as:

The three reserves are:

The IMO Team Leader is Dr Geoff Smith of the University of Bath, the Deputy Leader is Dr Ceri Fiddes of Stowe School, the Observer with Leader is Dr Joseph Myers of CodeSourcery and the Observer with Contestants is Pam Hunt.

The team leave the UK for pre-IMO training on 8 July, travel to Hanoi on 23 July and return to the UK on 1 August.

Balkan Mathematical Olympiad results (30 April 2007)

The UK team in the Balkan Mathematical Olympiad on Rhodes have achieved the following scores (each question marked out of 10).

The medal cut-offs were 33 for gold, 23 for silver and 15 for bronze. The problems are available on Mathlinks.

Balkan Mathematical Olympiad team selected (2 April 2007)

The UK has been invited as a guest nation to send a team to the Balkan Mathematical Olympiad, to be held on Rhodes, Greece from 26 April–2 May. The Team Leader is Robin Bhattacharyya of Highgate School, and the Deputy Leader is Dr Vesna Kadelburg of Sevenoaks School. After the training camp held at Trinity College, Cambridge from 29 March–2 April, the team has been selected as:

IMO squad selected (2 April 2007)

After the training camp held at Trinity College, Cambridge from 29 March–2 April, the squad of nine from which the team of six and three reserves for the IMO in Hanoi, Vietnam will be selected has been chosen as:

The Team Leader is Dr Geoff Smith of the University of Bath and the Deputy Leader is Dr Ceri Fiddes of Stowe School. The IMO will take place from 19–31 July.

BMO Round 2 top scorers (11 February 2007)

The following candidates scored 20 or more out of 40 in BMO Round 2. Invitations to the Trinity training camp will be sent shortly to selected candidates eligible to represent the UK at the IMO, and details of marks will be sent to schools in the week beginning 19 February.

BMO Round 2 marked (10 February 2007)

BMO Round 2 was marked in Bath on 10 February; invitations to the Trinity training camp will be sent shortly, and marks will be sent to schools in the week beginning 19 February.

BMO Round 2 (30 January 2007)

This year’s BMO Round 2 paper was taken today.

BMO Round 2 cut-offs (14 December 2006)

The BMO2 qualifying cut-off scores are 32 for students in year 13, 22 for students in year 12 and 18 for students in year 11 and below. Scoring at least the cut-off on BMO1 gives free entry to BMO2 provided that the candidate is eligible to represent the UK at the IMO. Any participant in BMO1 may enter BMO2 by paying the entry fee of £20. BMO2 will take place on 30 January 2007.

BMO Round 1 marked (8–10 December 2006)

BMO Round 1 was marked in Cambridge on 8–10 December; marks and information about Round 2 will be sent to schools shortly.

BMO Round 1 (1 December 2006)

This year’s BMO Round 1 paper was taken today.

Royal Society celebration (25 September 2006)

The IMO team’s success was celebrated at the Royal Society on Monday 25 September; Professor Sir Christopher Zeeman, FRS gave the 29th annual IMO lecture, on “Gyroscopes and Boomerangs”.

IMO 2006 report available (24 August 2006)

The reports on past IMOs now include Geoff Smith’s report on IMO 2006. The team’s success will be celebrated at the Royal Society on Monday 25 September; Professor Sir Christopher Zeeman, FRS will give the 29th annual IMO lecture, on “Gyroscopes and Boomerangs”.

IMO 2006 results (14–15 July 2006)

The IMO papers were sat on 12 and 13 July. The results of the UK team members are shown below. We came 19th out of 90 countries competing with four Silver and one Bronze medals and a score of 117 out of 252. The medal boundaries were 28 for Gold, 19 for Silver and 15 for Bronze.

IMO Paper 2 (13 July 2006)

Problem 4. Determine all pairs (x,y) of integers such that

1 + 2^x + 2^2x+1 = y².

(USA)

Problem 5. Let P(x) be a polynomial of degree n > 1 with integer coefficients and let k be a positive integer. Consider the polynomial Q(x) = P(P(…P(P(x))…)), where P occurs k times. Prove that there are at most n integers t such that Q(t) = t.

(Romania)

Problem 6. Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.

(Serbia and Montenegro)

Time allowed: 4 hours 30 minutes
Each problem is worth 7 points

IMO Paper 1 (12 July 2006)

Problem 1. Let ABC be a triangle with incentre I. A point P in the interior of the triangle satisfies

∠PBA + ∠PCA = ∠PBC + ∠PCB.

Show that AP ≥ AI, and that equality holds if and only if P = I.

(Korea)

Problem 2. Let P be a regular 2006-gon. A diagonal of P is called good if its endpoints divide the boundary of P into two parts, each composed of an odd number of sides of P. The sides of P are also called good.

Suppose P has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of P. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

(Serbia and Montenegro)

Problem 3. Determine the least real number M such that the inequality

|ab(a²−b²) + bc(b²−c²) + ca(c²−a²)| ≤ M(a²+b²+c²)²

holds for all real numbers a, b and c.

(Ireland)

Time allowed: 4 hours 30 minutes
Each problem is worth 7 points

IMO team in Ljubljana (10 July 2006)

After a pre-IMO training camp held jointly with the Slovenian team in Bled from 1–10 July, the UK team has now arrived at the IMO in Ljubljana. They will sit the papers on 12 and 13 July.

IMO team selected (31 May 2006)

After the training camp held at Oundle School from 27–31 May 2006, the team of six for the IMO (to be held in Ljubljana, Slovenia from 6–18 July) has been chosen as:

The two reserves are:

The IMO Team Leader is Dr Geoff Smith of the University of Bath, the Deputy Leader is Dr Ceri Fiddes of Millfield School and the Observer with Leader is Dr Joseph Myers of CodeSourcery.

The team leave the UK for pre-IMO training on 1 July, travel to Ljubljana on 10 July and return to the UK on 19 July.

Balkan Mathematical Olympiad detailed results (4 May 2006)

The UK team in the Balkan Mathematical Olympiad in Cyprus have achieved the following scores (each question marked out of 10).

The medal cut-offs were 32 for gold, 20 for silver and 11 for bronze. The problems are available on Mathlinks.

Balkan Mathematical Olympiad results and revised IMO squad (30 April 2006)

The UK team in the Balkan Mathematical Olympiad in Cyprus have achieved the following scores (out of 40).

The Team Leader at the Balkan Mathematical Olympiad is Richard Atkins of Oundle School, UK IMO Deputy Leader from 1999 to 2003, and the Deputy Leader is Dr Ian Jackson of Tonbridge School, UK IMO silver medallist in 1981.

In the light of his performance in the Balkan Mathematical Olympiad, Daniel has been added to the squad of eight from which the team of six and two reserves for the IMO in Ljubljana, Slovenia will be selected:

(Two of the eight students originally selected after the training camp held at Trinity College, Cambridge from 6–10 April 2006 had withdrawn in order to represent the UK in the Chemistry and Physics Olympiads.)

The IMO Team Leader is Dr Geoff Smith of the University of Bath and the Deputy Leader is Dr Ceri Fiddes of Millfield School. The IMO will take place from 6–18 July.

IMO squad selected (11 April 2006, revised 14–18 April 2006)

After the training camp held at Trinity College, Cambridge from 6–10 April 2006, the squad of seven from which the team of six and one reserve for the IMO in Ljubljana, Slovenia will be selected has been chosen as:

(Some students originally selected have withdrawn in order to compete in the Chemistry and Physics Olympiads.)

The Leader is Dr Geoff Smith of the University of Bath and the Deputy Leader is Dr Ceri Fiddes of Millfield School. The IMO will take place from 6–18 July.

Balkan Mathematical Olympiad team selected (5 March 2006)

The UK has been invited as a guest nation to send a team to the Balkan Mathematical Olympiad, to be held in Agros, Cyprus from 27 April to 3 May 2006. The Team Leader will be Richard Atkins of Oundle School, UK IMO Deputy Leader from 1999 to 2003, and the Deputy Leader will be Dr Ian Jackson of Tonbridge School, UK IMO silver medallist in 1981. The team has been selected as:

BMO Round 2 (31 January 2006)

This year’s BMO Round 2 paper was taken today.

BMO Round 1 prizewinners (19 January 2006)

The list of prizewinners at BMO1 is now available.

BMO Round 1 (30 November 2005)

This year’s BMO Round 1 paper was taken today.

Royal Society celebration (19 September 2005)

The IMO team’s success was celebrated at the Royal Society on Monday 19 September; Professor Frances Kirwan, FRS gave the 28th annual IMO lecture, on “Topology, Geometry and Strings”.

Balkan Mathematical Olympiad report available (16 September 2005)

The Balkan Mathematical Olympiad information now includes Robin Bhattacharyya’s report on the 2005 olympiad. Some photographs from that olympiad taken by Robin Bhattacharyya and Amanda Turner are also available online.

IMO 2005 report available (29 July 2005)

The reports on past IMOs now include Geoff Smith’s report on IMO 2005.

IMO 2005 results (16–20 July 2005)

The IMO papers were sat on 13 and 14 July. The results of the UK team members are shown below. We came 13th out of 91 countries competing with one Gold, three Silver and two Bronze medals and a score of 159 out of 252. This score of 159 is our highest since 1996, as is our total of four Gold or Silver medals. Martin Orr is the first ever IMO Gold medallist from Northern Ireland. The medal boundaries were 35 for gold, 23 for silver and 12 for bronze. The team have now returned to the UK, and their success will be celebrated at the Royal Society on Monday 19 September; Professor Frances Kirwan, FRS will give the 28th annual IMO lecture, on “Topology, Geometry and Strings”. See also Adrian’s last bulletin below.

Bulletin 2 from Adrian (16:14 BST, 20 July 2005)

Dear all,

We are now returned from Mérida, safe and well. I’m sorry that we were not able to keep in touch with you as much as usual during the IMO—computers were scarce, and our lives were further complicated during the last couple of days of the IMO by the arrival of Hurricane Emily.

The IMO papers and full results are available on the IMO website at www.imo2005.org

The UK came 13th with 1 gold, 3 silver and 2 bronze medals, a very good performance indeed for us. I shall not comment on the problems, since I imagine that many of you may still want to have a go at them.

For the first time in ten years, a special prize was awarded this year, to a student from Moldova who found an extremely quick and elegant solution to problem 3.

IMO 2005 will go down as one of the best ever. The Mexicans were warm and generous hosts, and very well organized throughout. The accommodation was excellent. So too was the co-ordination of the exam scripts, which is always a (the?) make-or-break test for a good IMO.

Many thanks to Joseph for the excellent support that he provided back in the UK, and thanks once again to all of you for all your brilliant assistance with UK IMO preparation this year.

Best wishes, Adrian

(Full text of message available.)

Hurricane Emily (18 July 2005)

Hurricane Emily has now passed by Mérida; all IMO participants are safe. There is no current indication of any disruption to the UK team’s return journey, although the UK IMO leadership in Mérida have not yet definitely confirmed that the flights are on as scheduled.

Hurricane Emily (17 July 2005)

Hurricane Emily is expected to pass over Mérida on Monday. IMO participants are safely sheltered in windowless safe rooms in the IMO hotels which are the strongest buildings in the area and serve as evacuation centres for the surrounding area. Further news will be added to this site when available, but the hurricane may bring down communications links from Mérida.

IMO Paper 2 (14 July 2005)

Problem 4. Consider the sequence a₁, a₂, ... defined by

a_n = 2ⁿ + 3ⁿ + 6ⁿ - 1  (n = 1, 2, ...).

Determine all positive integers that are relatively prime to every term of the sequence.

Problem 5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The lines AC and BD meet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R. Consider all the triangles PQR as E and F vary. Show that the circumcircles of these triangles have a common point other than P.

Problem 6. In a mathematical competition 6 problems were posed to the contestants. Each pair of problems was solved by more than 2/5 of the contestants. Nobody solved all the 6 problems. Show that there were at least 2 contestants who each solved exactly 5 problems.

IMO Paper 1 (13 July 2005)

Problem 1. Six points are chosen on the sides of an equilateral triangle ABC: A₁, A₂ on BC; B₁, B₂ on CA; C₁, C₂ on AB. These points are the vertices of a convex hexagon A₁A₂B₁B₂C₁C₂ with equal side lengths. Prove that the lines A₁B₂, B₁C₂ and C₁A₂ are concurrent.

Problem 2. Let a₁, a₂, ... be a sequence of integers with infinitely many positive terms and infinitely many negative terms. Suppose that for each positive integer n, the numbers a₁, a₂, ..., a_n leave n different remainders on division by n. Prove that each integer occurs exactly once in the sequence.

Problem 3. Let x, y and z be positive real numbers such that xyz ≥ 1. Prove that (x⁵-x²)/(x⁵+y²+z²) + (y⁵-y²)/(y⁵+z²+x²) + (z⁵-z²)/(z⁵+x²+y²) ≥ 0.

Bulletin 1 from Adrian (07:53 BST, 2 July 2005)

All being well, we shall leave for our final IMO preparation camp in Houston this morning. The team is in very good shape—thanks in large part to the assistance that you have all given them over the past months. So I hope that we shall have good news to report from Mexico in a couple of weeks time. Good news or not so good, I’ll keep you informed of our progress so far as I can. The IMO papers are on the 13th and 14th, so complete results should be known by the evening of the 16th. Please do forward any information that I send to anyone else who you think might be interested.

Adrian

(Full text of message available.)

IMO team selected (1 June 2005)

After the training camp held at Oundle School from 28 May–1 June 2005, the team of six for the IMO (to be held in Mérida, Yucatán, México from 8–19 July 2005) has been chosen as:

The two reserves are (in alphabetical order):

This was a particularly difficult decision, and the selection committee spent a very long time trying to discriminate between the sixth and seventh placed students. Their work was examined in considerable detail.

The team leave the UK for pre-IMO training on 2 July, travel to Mérida on 11 July and return to the UK on 20 July.

Balkan Mathematical Olympiad results (7–8 May 2005)

The UK team in the Balkan Mathematical Olympiad have achieved the following scores. Each question is marked out of ten; the problems are available on the Balkan Mathematical Olympiad website. The medal cutoffs are 33 for gold, 22 for silver and 10 for bronze.

The UK placed 11th out of 14 teams at the olympiad:

189 Romania
146 Bulgaria
139 Moldova
134 Kazakhstan
128 Romania B
127 Turkey
115 Hungary
99 Greece, Serbia and Montenegro
71 Albania
54 UK
48 Macedonia
30 Yakutia
23 Cyprus

IMO squad selected (12 April 2005)

After the training camp held at Trinity College, Cambridge from 7–11 April 2005, the squad of eight from which the team of six and two reserves for the IMO in Mérida will be selected has been chosen as:

Balkan Mathematical Olympiad team selected (24 February 2005)

The UK team for the Balkan Mathematical Olympiad in Iași, Romania has been chosen as:

Balkan Mathematical Olympiad arrangements (25 January 2005)

Robin Bhattacharyya will lead the UK team to the Balkan Mathematical Olympiad in Iași, Romania during May 4–10, 2005. Robin is a teacher at Highgate School. (Robin was a reserve for the UK IMO team in 1992.)

Amanda Turner, a research student at the University of Cambridge, will be the deputy leader. She represented South Africa at IMOs in 1996 and 1997. At the second event she was awarded a bronze medal.

The team will be selected at the BMO 2 marking session on February 12th. A public announcement of the team is expected by February 19th.

BMO Round 1 results histogram available (17 January 2005)

A histogram of distribution of marks in BMO Round 1 is now available.

BMO Round 1 marked (13 December 2004)

The BMO 1 scripts were marked in Peterborough on 10–11 December. The marks and BMO 2 invitations will be sent to schools shortly.

IMO 2005 moved from Cancún to Mérida (25 November 2004)

IMO 2005, originally proposed to be held in Cancún, Quintana Roo, México, will now instead be held in Mérida, Yucatán, México from July 8–19, 2005, a week later than originally planned, to avoid possible impact from any political changes arising from state elections to be held in Quintana Roo before July 2005.

Ceri Fiddes to be next IMO Deputy Leader (10 November 2004)

Dr Ceri Fiddes, a teacher at Millfield School in Somerset, will be the UK Deputy Leader from IMO 2006 (Slovenia) onwards, the first female UK IMO Leader or Deputy Leader since 1972.

Royal Society celebration (20 September 2004)

The team’s success was celebrated at the Royal Society on Monday 20 September; Professor Tim Gowers, FRS gave the 27th annual IMO lecture, on “How to specify very very large integers”.

IMO 2004 report available (28 July 2004)

The reports on past IMOs now include Geoff Smith’s report on IMO 2004.

IMO 2004 results (15–19 July 2004)

The IMO papers were sat on 12 and 13 July. The results of the UK team members are shown below. We came 20th out of 85 countries competing with one Gold, one Silver and four Bronze medals and a score of 134 out of 252. For individual bulletins from Greece, see below. See also the UKMT press release.

Bulletin 3 from Adrian (07:15 BST, 16 July 2004)

Here is some further information on the IMO—mostly for Joseph’s benefit, but it may be of interest to some others of you as well.

The full UK results were

UNK1 (Coope) 7 2 2 7 3 2 = 23
UNK2 (Fidler) 7 6 1 7 3 0 = 24
UNK3 (Jefferys) 7 7 2 7 3 6 = 32
UNK4 (Orr) 6 7 2 7 0 0 = 22
UNK5 (Shannon) 2 6 0 7 1 0 = 16
UNK6 (Shepherd) 7 1 1 7 1 0 = 17

There was some beautiful stuff from David Fidler. As last year, we were reasonably efficient on the easier questions 1,2,4—but rather disappointing on the rest, particularly question 5.

There were 87 countries participating altogether, including Mozambique (MOZ) and Saudi Arabia (KSA) for the first time. The number of Arab nations was higher than usual—Morocco, Tunisia, Kuwait, Saudi Arabia.

Forthcoming hosts were announced: Spain 2008 (unknown location), Germany 2009 (Bremen).

Full results for countries were as follows:

Albania 57
Argentina 92
Armenia 98
Australia 125
Austria 55
Azerbaijan 72
Belarus 154
Belgium 86
Bosnia and Hercegovina 40
Brazil 132
Bulgaria 194
Canada 132
China 220
Colombia 122
Croatia 89
Cuba 17 (one contestant)
Cyprus 49
Czech Rep 109
Denmark 46
Ecuador 14
Estonia 85
Finland 49
FYROM 71
France 94
Georgia 123
Germany 130
Greece 126
Hong Kong 120
Hungary 187
Iceland 35
India 151
Indonesia 61
Iran 178
Ireland 48
Israel 147
Italy 69
Japan 182
Kazakhstan 132
Korea 166
Kuwait 5
Kyrgystan 63
Latvia 63
Lithuania 65
Luxembourg 36 (3 contestants)
Macau 86
Malaysia 34
Mexico 96
Moldova 140
Mongolia 135
Morocco 88
Mozambique 13 (3)
Netherlands 53
New Zealand 56
Norway 55
Paraguay 13 (3)
Peru 49 (3)
Phillipines 16 (5)
Poland 142
Portugal 26
Puerto Rico 43 (5)
Romania 176
Russia 205
Saudi Arabia 4
Serbia and Montenegro 132
Singapore 139
Slovakia 119
Slovenia 69
South Africa 110
Spain 57
Sri Lanka 33
Sweden 75
Switzerland 57
Taiwan 190
Thailand 99
Trinidad and Tobago 29 (5)
Tunisia 31
Turkey 119
Turkmenistan 52
Ukraine 174
UK 134
USA 212
Uruguay 47
Uzbekistan 79
Venezuela 15 (2)
Vietnam 196

Apologies for spelling errors in the above: some have crept in, I’m sure.

UK was third in the EU (Hungary, Poland) and third in the Commonwealth (India, Singapore).

The four students with 42 came from Canada, Hungary, Russia and Russia.

The UK students are all delighted with their medals, and are now enjoying the post-exam part of the IMO.

Adrian + Geoff

(Full text of message available.)

Bulletin 2 from Adrian (23:36 BST, 15 July 2004)

Well everything is now known. The UK points totals were as follows:

Giles Coope  23 bronze
David Fidler  24 silver
Paul Jefferys  32 gold
Martin Orr  22 bronze
Alexander Shannon  16 bronze
Anne Marie Shepherd  17 bronze

These marks were fair. I think some of our students might have done a little better on another day, but getting 6 medals is great. We got full marks on q.4, and solid totals on q.1 and q.2, but rather little elsewhere.

The cutoffs were 16, 24 and 32. We came 19th. The top four teams in order were China, USA, Russia and Vietnam. Four students got 42s.

I shall provide more complete data (as requested by Joseph) tomorrow morning on our performance and on the IMO scores in general. I just need to find Gordon Lessells...

Many of you have emailed me interesting comments about the papers—sorry I haven’t had the chance to reply to you individually. I think the consensus around here is that the papers were pretty good. Question 3 was found relatively difficult (< 10 solvers, I think), and q.6 relatively straightforward. I suspect when the data is analysed that q.5 will be found to be a rather difficult one for that position on the paper.

The IMO experience as a whole has been excellent, though a somewhat improvisatory sense of organization has prevailed at times.

Best wishes to one and all,

Adrian

(Full text of message available.)

Bulletin 1 from Adrian (05:06 BST, 14 July 2004)

Coordination begins this morning, and I should be able to give bulletins from now on. I won’t say anything yet about how our team and others have found the problems, because I imagine many of you will want to have a go at them.

We have performed solidly—and in case you are wondering, Paul does not have a 42 possibility. Six medals looks realistic for us at this stage.

Best wishes,

Adrian

(Full text of message available.)

Papers 1 and 2 (12 and 13 July 2004)

These were formerly provided here; they are now included in Geoff Smith’s report on IMO 2004, available from the collection of reports on past IMOs.

IMO 2004 papers and results (11 July 2004)

The IMO papers are being sat on 12 and 13 July, and we wish the UK team the best of luck in those papers, and our Leader, Deputy Leader and Observers the best of luck in the subsequent Co-ordination. Results will appear here as soon as they are available, probably on 15 or 16 July.

IMO team selected (2 June 2004)

The UK team for IMO 2004 in Athens has been chosen as:

The three reserves are:

IMO squad selected (5 April 2004, school details added 6 April 2004)

After the training camp held at Trinity College, Cambridge from 1–5 April 2004, the squad of nine from which the team of six and three reserves for the IMO in Athens will be selected has been chosen as:

BMO Round 2 (24 February 2004)

This year’s BMO Round 2 paper was taken today.

BMO Round 1 (3 December 2003)

This year’s BMO Round 1 paper was taken today.

Royal Society Celebration (29 September 2003)

The team’s success was celebrated at the Royal Society on Monday 29 September; Professor Adrian Smith FRS gave the 26th annual IMO lecture on “Why Detectives and Judges need to understand probability”.

IMO 2003 report available (31 July 2003)

The reports on past IMOs now include Geoff Smith’s report on IMO 2003.

UK IMO results 2003 (17 July 2003, last updated 21 July 2003)

In IMO 2003 held in Tokyo, the UK came equal 10th (with Hungary), with one Gold, two Silver and three Bronze medals, being the top Western European team and the top Commonwealth team. Individual scores (all in the top 30% of scores) were: