## Dec 16, 2012

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### day 1

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Language: English. Problem 1. Given a finite number of boys and girls, a sociable set of boys is a set of boys such that every girl knows at least one boy in that ...
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### day 2

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THE 3RD ROMANIAN MASTER OF MATHEMATICS COMPETITION. DAY 2: SATURDAY, FEBRUARY 27, 2010, BUCHAREST. Language: English. Problem 4.
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### day 1

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THE 3RD ROMANIAN MASTER OF MATHEMATICS COMPETITION. DAY 1: FRIDAY, FEBRUARY 26, 2010, BUCHAREST. Language: English. Problem 1.
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### day 2

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Language: English. Problem 4. Prove that there are infinitely many positive integers n such that 22n+1 + 1 is divisible by n but 2n + 1 is not. (Russia) Valery ...
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### original

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Problem 1. Let ABC be an equilateral triangle. P is a variable point internal to the triangle and its perpendicular distances to the sides are de- noted by a2, b2 ...
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### Problem - British Mathematical Olympiad

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Language: English. Day: 1. Monday, July 18, 2011. Problem 1. Given any set A = {a1,a2,a3,a4} of four distinct positive integers, we denote the sum a1 +a2 +a3 ...
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### day 1

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THE 4th ROMANIAN MASTER OF MATHEMATICS COMPETITION. DAY 1: FRIDAY, FEBRUARY 25, 2011, BUCHAREST. Language: English. Problem 1.
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### 24 February 1994 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Thursday, 24 February 1994. Time allowed Three and a half hours. Each question is worth 10 marks.
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### FST 2, 11 April 2005

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UK IMO FST2. Trinity College, Cambridge. 11 April 2005. 1. The circle Γ and the line l do not intersect. Let AB be the diameter of. Γ which is perpendicular to l, ...
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### FST, 2 April 1995

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FlNAL SELECTION TEST. SUNDAY 2 APRlL1995. WM: W W M Mew W W, WWW. 7%! W nwu/\12.a/_ \ W M W 06 M“ W,. L \ Wdrwl Z [4!/'CAO M,. 2. A positive ...
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### Next Selection Test: Paper 3

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Next Selection Test: Paper 3. Oundle School. 31st May 2011. 1. If X is a set of integers, define D(X) to be the set of differences between elements of X: D(X) = {n ...
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### Problem 1. For positive integers a1,...,ak, let n = ∑ ai , and let ( n a1 ...

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Saturday, February 28, 2009, Bucharest. Language: English. Problem 1. For positive integers a1,...,ak, let n = k. ∑ i=1 ai , and let. ( n a1,...,ak. ) be ...
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### 13 January 1999 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday, 13 January 1999. Time allowed Three and a half hours. Instructions • Full written solutions - not ...
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### 24 March 1976 - British Mathematical Olympiad

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NATIONAL COMMlTTEE FOR MATHEMATICAL CONTESTS. British Mathematical Olympiad. 1976. Time allowed ~ 3% hours. 24th March, 1976. Write on one ...
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### 2009 United Kingdom & Australia Pre-IMO Camp Trinity College ...

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2009 United Kingdom & Australia Pre-IMO Camp. Trinity College, Cambridge. 2nd Test. Thursday 9 July. The Ashes. • Each question is worth 7 points.
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### FST, 13 April 1997

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FINAL SELECTION TEST. SUNDAY 13 APRIL 1997. Time allowed: 4% hours. A ﬁnite sequence of integers a0,a1, ...,a,l is called quadratic if. |a,- — ai_1I = i2 for ...
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### FST 1, 5 April 2008

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First Selection Test: Exam 1. IMO camp, Trinity College Cambridge. 5—iv-2008. Problem 1 A triangle ABC is given. A circle F passes through A and is tangent to ...
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### NST 1, 27 May 2007

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Oundle Test 1. 27 May 2007. 1. Triangle ABC has circumcentre O and centroid M. The lines OM and AM are perpendicular. Let AM meet the circumcircle of ABC ...
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### 5 December 2001 - British Mathematical Olympiad

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British Mathematical Olympiad. Round 1 : Wednesday, 5 December 2001. Time allowed Three and a half hours. Instructions • Full written solutions - not just ...
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### 11 May 1983

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G1. I-'urthor Inter-national Selection Test. Wednesday May 11, 1933. 'i'i1ne allowed - 5! hours. Wrrte your name on each guge. Start each question on G new ...
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### 1 March 1989

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NATIQLAL mm H12 HATHE'iATICAL ms. Further International Selection Test 1989. Weclnesda;vP lat March 1989. Time allowed: 3% HOURS. PLEASE READ ...
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### 28 February 1991

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FURTHER lNTERNATlONAL SELECTION TEST 1991. Thursday 28th February 1991. Time allowed: 3'/2 hours. - Start each question on a fresh sheet of paper.
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### 5 February 1988

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5 Feb 1988 – NATIONAL OOMMI'I'I'EE FOR MATHEMATICAL CONTESTS. Further International Selection Test. Friday 5th February 1988. Time allowed ...
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### 17 March 1977 - British Mathematical Olympiad

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NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. British Mathematical Olympiad. 17th March, 1977. Time allowed -' 3i hours. Write on one side of ...
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### 27 February 2001 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Tuesday, 27 February 2001. Time allowed Three and a half hours. Each question is worth 10 marks.
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### NST 4, 2 June 2010

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2 Jun 2010 – NST 4. 2 June 2010. 1. Consider the sequence (ai) such that a0 = 4, a1 = 22 and an − 6an−1 + an−2 = 0 for n ≥ 2. Prove that there are integral ...
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### 1973 - British Mathematical Olympiad

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BRITISH MATHEMATICAL. OLYIVIPIAD. 1973. TIME: 3 HOURS. PLEASE NOTE INVTGILATOR'S INSTRUCTIONS. Two fixed circles are touched by a Variable ...
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### 5 May 1976

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NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. Further International Selection Test, 1976. May 5thI 3! hours. Through a point P in the interior of a ...
29. [PDF]

### 5 March 1985 - British Mathematical Olympiad

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NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. British Mathematical Olympiad. Tuesday 5 March, 1985. Time allowed - 3% hours. Write on one ...
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### 15 March 1985 - British Mathematical Olympiad

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NATIGNAL COMMITTEE FOR MATHEMATICAL CONTESTS. Further International Selection Test. Friday, March 15th 1985. Time allowed - 3 hours. Write on one ...
31. [PDF]

### Next Selection Test: Paper 4

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Next Selection Test: Paper 4. Oundle School. 1st June 2011. 1. Let A be the set of all integers of the form a2 + 13b2, where a and b are integers and b is nonzero ...
32. [PDF]

### FST 1, 3 April 2004

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First Selection Test 1. 3-iv-2004. 1. Three distinct points A, B and C are fixed on a line in that order. Let. Γ be a circle passing through A and C whose centre does ...
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### BRITISH MATHEMATICAL OLYMPIAD 1974 Time allowed: 3 hours

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BRITISH MATHEMATICAL OLYMPIAD 1974. Time allowed: 3 hours. Each question should be answered on a fresh sheet of paper. Use one side of the paper ...
34. [PDF]

### FST 2 2010

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FST 2 2010. Trinity College, Cambridge. 12th April 2010. 4. Find all solutions to p(p + 1) + q(q + 1) = n(n + 1) where p and q are prime numbers and n is a ...
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### SIST, 16 April 1989

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NATIGJAL (I1'MI'I'I'EE FCR MATHEMTICAL (XNTBSTS. Second International Selection Test. Cambridge, 16th April 1989. Time allowed 3% hours. Please write ...
36. [PDF]

### NST 3, 5 June 2012

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Next Selection Test: Paper 3. Oundle School, Northamptonshire. 5th June 2012. 1. Graphistan has 2011 cities; the company Graph Air sells flights in one ...
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### NST3 2009

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NST3 2009. Eton May 21, Oundle May 26. 1. Let ABC be an acute-angled triangle, and M be a point in its plane distinct from the vertices Show that the vector ...
38. [PDF]

### 1993-2 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Thursday, 11 February 1993. Time allowed Three and a half hours. Each question is worth 10 marks.
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### 1974

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'ﬁaﬁzher Internati.czzal. Selectisn 'fest, 191%. Time 3 hears. Ne tables and n0 lists of formulae are alleweﬁ. 'i. {i} In AABC sin2A + sin3B + sinzﬁ = 2. Prove that ...
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### 1994-1 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday 19th January 1994. Time allowed Three and a half hours. Instructions • Full written solutions are ...
41. [PDF]

### 13 December 1988 - British Mathematical Olympiad

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13 Dec 1988 – NATIONAL COMl\/IITTEE FOR MATIIEMATICAL CONTESTS. BRITISH MATHEMATICAL OLYMPIAD. Tuesday 13th December 1988 ...
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### British Mathematical Olympiad Round 1 2008/09 Masterclass

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6 Dec 2008 – UNITED KINGDOM MATHEMATICS TRUST. British Mathematical Olympiad Round 1. 2008/09 Masterclass. 10 am – 12 noon, Saturday 6 ...
43. [PDF]

### 25 February 1999 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Thursday, 25 February 1999. Time allowed Three and a half hours. Each question is worth 10 marks.
44. [PDF]

### 14 May 1981

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14 May 1981 – NATIONAL COMMITTEE I08 IATBEIATICAL OONTISTB. Further international Selection Teet. 14th May 1981 Tine allowed - 3} houre. Start each ...
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### FST, 18 April 1993 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD COMMITTEE. FINAL SELECTION TEST. Sunday 18th April 1993. Time allowed : 4% hours. Brummie dwarves are ...
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### FST, 5 April 1992 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD COMMITTEE. FINAL SELECTION TEST. Sunday 5th April 1992. Time allowed : 4% hours. 1. Circles C1, C2, with centres ...
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### PRESS RELEASE UK MASTERS MATHEMATICS FOR MASTERS ...

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13 Feb 2008 – UNITED KINGDOM MATHEMATICS TRUST. The UKMT is a company limited by guarantee and registered in England and Wales. Registered ...
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### NST, 5 June 2002

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Next Selection Test: 4 hours 30 minutes. Oundle, June 5, 2002. 1. Let ABC be a triangle and l the line through C which is parallel to AB r. The internal bisector of ...
49. [PDF]

### 17 May 1972

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Further International Selection rBest. May 17, 1972. Time 5 hours. Only your best 5 answers will score°. 1 Show 1101'? to assigli to the vertices of a regular ...
50. [PDF]

### AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE 2010 IMO ...

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AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE. 2010 IMO Team Training. Exam T16. • Each question is worth 7 points. • Time allowed is 41. 2 hours.
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### NST 3, 27 May 2008

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Next Selection Test: Exam 3. IMO camp, Oundle School. 2"/'~v-2008. Problem 7 Find all injective functions f 1 N -——+ N such that, for each n,. ' "l- f<r<~>> g ...
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### FST2 2006

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FST2 2006. Trinity College Cambridge. 1. Each integer is coloured either red, blue, green or white. Let :1; and y be odd integers such that 75 Show that there ...
53. [PDF]

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THE MATHEMATICAL ASSOCIATION. 259 LONDON ROAD LEICESTER LE2 38E. 0533 703877. NATIONAL COMMITTEE FOR. MATHEMATICAL CONTESTS' ...
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### 10 March 1983 - British Mathematical Olympiad

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5/. THE MATHEMATICAL ASSOCIATION. National Committee for Mathematical Contests. British Mathematical Olympiad. 10th March 1983. Time allowed — 3% ...
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### FST, 10 April 1994 - British Mathematical Olympiad

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BRITISH MATHEMATICAL OLYMPIAD COMMITTEE. FINAL SELECTION TEST. Sunday 10th April 1994. Time allowed : 4% hours. Prove that if a, b, c are ...
56. [PDF]

### NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS - British ...

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NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. British Mathematical Olympiad. Friday 20th March 1987. Time allowed — 3% hours. PLEASE ...
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### FST 1, 31 March 2007

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First Selection Test: Exam 1. 31-iii~2007. Problem 1 Let TL be a natural number. VVe want to colour each natural number red or blue so that the following ...