BMO compilation (I)
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Quick View BMO Marking. Each question on the British Mathematical Olympiad is marked out of a total of. 10 marks. The style of marking is rather different to that which most ...
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Quick View Areal Co-ordinate Methods in Euclidean Geometry. Tom Lovering. April 11, 2008. Introduction. In this article I aim to briefly develop the theory of areal (or ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday 13th January 1993. Time allowed Three and a half hours. Instructions • Full written solutions are ...
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Quick View NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. BRITISH MATHEMATlCAL OLYMPIAD. Wednesday 17th January 1990. Time allowed - Three and ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD, 1975. 24th March, 1975. Time allowed - 3 hours. Write on one side of the paper only. Start each question on a fresh ...
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Quick View 30 Nov 2007 – United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 1 : Friday, 30 November 2007. Time allowed 31. 2 hours.
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Quick View 2 Dec 2010 – United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 1 : Thursday, 2 December 2010. Time allowed 31. 2 hours.
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Quick View Supported by. British Mathematical Olympiad. Round 2 : Tuesday, 25 February 2003. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View Language: English. Day: 1. Tuesday, July 10, 2012. Problem 1. Given triangle ABC the point J is the centre of the excircle opposite the vertex A. This excircle is ...
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Quick View United Kingdom Mathematics Trust. UK Mathematical Olympiad for Girls. 20 September 2012. Instructions. • Time allowed: 3 hours. • Full written solutions – not ...
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Quick View BRlTlSH MATHEMATICAL OLYMPIAD. Round 2 : Thursday 13th February 1992. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View 2 Dec 2011 – United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 1 : Friday, 2 December 2011. Time allowed 31. 2 hours. Instructions ...
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Quick View Supported by. British Mathematical Olympiad. Round 1 : Wednesday, 3 December 2003. Time allowed Three and a half hours. Instructions • Full written ...
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Quick View 1 Dec 2006 – UKMT Maths Challenges Office, School of Mathematics, University of Leeds, Leeds LS2 9JT. Tel: 0113 343 2339 Fax: 0113 343 5500 Email: ...
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Quick View Supported by. British Mathematical Olympiad. Round 1 : Wednesday, 30 November 2005. Time allowed 31. 2 hours. Instructions • Full written solutions - not just ...
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Quick View Supported by. British Mathematical Olympiad. Round 1 : Wednesday, 1 December 2004. Time allowed Three and a half hours. Instructions • Full written ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday 15th January 1992. Time allowed. Instructions 0. Three and a half hours. Full wrttten solutions are ...
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Quick View 4 Dec 2008 – United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 1 : Thursday, 4 December 2008. Time allowed 31. 2 hours.
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Quick View European Girls' Mathematical Olympiad. EGMO | 2012. UK Mathematical Olympiad for Girls. 23 June 2011. Instructions. Supported by. • Time allowed: 3 hours.
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Quick View 10 Aug 2010 – 9th Chinese Girls' Mathematics Olympiad. Shijiazhuang, China. Day I. 8:00 AM - 12:00 PM. August 10, 2010. 1. Let n be an integer greater than ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday, 17 January 2001. Time allowed Three and a half hours. Instructions • Full written solutions - not ...
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Quick View Supported by. British Mathematical Olympiad. Round 2 : Tuesday, 1 February 2005. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Thursday, 15 February 1996. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 2 : Thursday, 28 January 2010. Time allowed Three and a half hours. Each question ...
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Quick View United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 2 : Thursday, 29 January 2009. Time allowed Three and a half hours. Each question ...
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Quick View Supported by. British Mathematical Olympiad. Round 2 : Tuesday, 30 January 2007. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Wednesday 16th January 1991. Time allowed - Three and a half hours. Instructions: '. Start each question on a fresh ...
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Quick View United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 2 : Thursday, 26 January 2012. Time allowed Three and a half hours. Each question ...
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Quick View United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 2 : Thursday, 31 January 2008. Time allowed Three and a half hours. Each question ...
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Quick View 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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Quick View 30 Nov 2012 – United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 1 : Friday, 30 November 2012. Time allowed 31. 2 hours.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Wednesday, 23 February 2000. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Thursday, 27 February 1997. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday 13th January 1993. Time allowed Three and a half hours. Instructions • Full written solutions are ...
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Quick View 3 Dec 2009 – United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 1 : Thursday, 3 December 2009. Time allowed 31. 2 hours.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday, 17th January 1996. Time allowed Three and a half hours. Instructions • Full written solutions - not ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 2 : Thursday, 26 February 1998. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View ( )0)1 ¥¨2 34¡¤§ 576869). 6A@ABC)EDGFH)E5IBQP"D. RTS UWVEX`Y% aQb' YcVedf bhgiXqprpQdsbtY4 uwvEg x8d¤y €ƒ‚ bh„GvEXTyhdHb a…
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Quick View British Mathematical Olympiad. Round 2 : Tuesday, 26 February 2002. Time allowed Three and a half hours. Each question is worth 10 marks. Instructions • Full ...
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Quick View United Kingdom Mathematics Trust. British Mathematical Olympiad. Round 2 : Thursday, 27 January 2011. Time allowed Three and a half hours. Each question ...
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Quick View Supported by. British Mathematical Olympiad. Round 1 : Friday, 1 December 2006. Time allowed 31. 2 hours. Instructions • Full written solutions - not just ...
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Quick View NATIONAL C0l'iHI.T'i'EE FOR 1'ง,ปง'}_" L'lEhAT ICAL CONTESTS. British Mathe.'.uatica,l 0lympia.d lbth March, 1979. Time allosed - 3% hours. Write on one ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday, 15 January 1997. Time allowed Three and a half hours. Instructions • Full written solutions - not ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday, 14 January 1998. Time allowed Three and a half hours. Instructions • Full written solutions - not ...
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Quick View Wednesday, July 15, 2009. Problem 1. Let n be a positive integer and let a1,...,ak (k ≥ 2) be distinct integers in the set. {1,...,n} such that n divides ai(ai+1 −1) for i ...
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Quick View FURTHER INTERNATIONAL SELECTION TEST. May 5th, 1975 3% hours. 1. In this question a "real function" means a function f such that f(x) exists and is real ...
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Quick View NATIONAL CUMMITTEE FOR MATHEMATICAL CGNTESTS. British Mathematical Olympiad. Tuesday 4th March 1986. Time allowed ~ 3% hours . Write on one ...
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Quick View NATIONAL COMMITTEE FOR MA'I'HEMATICAL CONTESTS. British Mathematical Olympiad. Friday 20th November 1987. Time allowed - 3% hours. PLEASE ...
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Quick View AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE. 2012 IMO Team Training. Exam T16. • Each question is worth 7 points. • Time allowed is 41. 2 hours.
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday 18th January 1995. Time allowed Three and a half hours. Instructions • Full written solutions are ...
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Quick View BRITISH MATHEMATICAL OLYMPIAD. Round 1 : Wednesday, 12 January 2000. Time allowed Three and a half hours. Instructions • Full written solutions - not ...
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Quick View 13 Mar 1980 – Write on one side of the paper only. fresh sheet of paper. NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. British Mathematical ...
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Quick View Supported by. British Mathematical Olympiad. Round 2 : Tuesday, 24 February 2004. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View Supported by. British Mathematical Olympiad. Round 1 : Wednesday, 11 December 2002. Time allowed Three and a half hours. Instructions • Full written ...
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Quick View 23 Mar 1984 – NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. Further International Selection Test. Friday, 23rd March 1984. Time allowed - 3} ...
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Quick View THE MATHEMATICAL ASSOCIATION. National Committee for Mathematical Contests. British Mathematical Olympiad. 18th March, 1982. Time allowed ~ 3% ...
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Quick View Supported by. British Mathematical Olympiad. Round 2 : Tuesday, 31 January 2006. Time allowed Three and a half hours. Each question is worth 10 marks.
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Quick View 23 Apr 1988 – National Committee for Mathematical Contests. Second International Selection Test. Reading, 23rd April 1988. Time allowed : 355 hours ...
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Quick View FIST 2, Bristol , May, 1985. Q1 Let x1,x2,...xn be real numbers such that oéxiéz for each 1. . Prove that. n n. 2: Z [X1-xi' \< 11*.' When does equality hold'? Q2 The ...
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Quick View NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. Second International Selection Test. Reading, Saturday 10th May 1.986. 3% hours. A, B, C, A ', B ' ...
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Quick View NATIONAL COMMITTEE FOR MATHEMATICAL CON"I'E.':1TS. Further International Selection Test , 19T8. May 12th 1918 - 3% hours. A plane convex pentagon ...
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Quick View AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE. 2011 IMO Team Training. Exam T14. • Each question is worth 7 points. • Time allowed is 41. 2 hours.
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Quick View First Selection Test. Trinity IMO Training Camp. 7 April 2002. 1. We are given a circle Γ and a straight line l which is tangent to the circle at the point B. Through ...
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Quick View 14 Apr 1991 – SECOND INTERNATIONAL SELECTION TEST. 'IYin.ity College, Cambridge, 14th April 1991. Time allowed: Three-and-a-half hours. Let f be ...
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Quick View First Selection Test. April 2003. 1. Consider triangle ABC. Let U,V,W be points such that U is on the line through B and C, V is on the line through C and A, W is ...
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Quick View First Selection Test: Paper 1. Trinity College, Cambridge. 16th April 2011. 1. Let ABC be an acute triangle with D, E, F the feet of the altitudes lying on BC, CA, AB ...
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Quick View AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE. 2008 IMO Team Training. Exam T15. • Each question is worth 7 points. • Time allowed is 41. 2 hours.
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Quick View 30. NATIONAL COMMITTEE FOR MATHEMATICAL CONTESTS. Further International Selection Test, 1912. May lOth 1979 - 3% hours. a, b, c , d are different ...
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Quick View 31 Mar 1996 – FINAL SELECTION TEST. SUNDAY 31 MARCH 1996. Time Allowed: 4% hours. 1. Two circles F1 and F2 intersect at D and E. The point B on ...
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Quick View 8 May 1980 – National Committee for Mathematical Contests. Further International Selection Test. May 8, 1980 5% hours. 1. VLMW and VABC are tetrahedra ...
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Quick View NATIONAL CXXMITFEE HR MA'!'HE4ATICAL CDNTESTS. Training Weekend 1989. GEIMETRY TEST. Tine allowed 1% hours. A, B are two points on a sphere ...
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Quick View 1 Jun 2010 – NST 3. 1 June 2010. 1. Let n ≥ 2 be an integer. At every point of the co-ordinate plane with integral co-ordinates (i, j) we write i + j modulo n (an ...
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Quick View Oundle Selection Test 1. 4 hours 30 minutes . Find all nondecreasing functions f : R -> R such that. (Ii) 1%) = 0) f(1) = 1;. (b) f(a) + f(b) : f(a)f(b) + f(a + b — ab) for ...
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Quick View Next Selection Test: Paper 2. Oundle School, Northamptonshire. 4th June 2012. 1. Let ABC be an acute triangle. Let ω be a circle whose centre L lies ...
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Quick View Next Selection Test: Exam 1. IMO carnp7 Oundle School. 25-v-2008. Problem 1 Let M and N be vertices of a cube. Assign the number 1 to these vertices and O ...
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Quick View 30 May 2010 – NST 1. 30 May 2010. 1. Consider a regular 2007-gon. Find the smallest positive integer k having the property that in every set of k vertices, ...
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Quick View UK IMO Next Selection Test 2. Oundle 2005. 1. Let n ≥ 2 be a natural number. A pyramid P has base A1A2 ···A2n and apex O. The polygon A1A2 ···A2n is ...
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Quick View 5 Apr 1998 – FINAL SELECTION TEST. SUNDAY 5 APRIL 1998. Time allowed: 4% hours. 1. Let P(:c) be a polynomial with real coefficients such that ...
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Quick View Next Selection Test: Paper 1. Oundle School, Northamptonshire. 3rd June 2012. 1. For any integer d > 0, let f(d) be the smallest positive integer that has exactly ...
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Quick View Oundle Test 3. 29 May 2007. 1. Let ABC be a triangle with LB 75 LC. The incircle I of ABC touches the sides BC, CA AB at the points D, E, F , respectively. Let AD ...
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Quick View Next Selection Test: Paper 2. Oundle School. 30th May 2011. 1. Find all functions f : Z → Z such that: (a) f(x + f(x + 2y)) = f(2x) + f(2y) for all integers x, y;. (b) f(0) ...
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Quick View NST 2, Eton and Oundle7 2009. 1. There are 2008 red cards and 2008 white cards. They are shufiled and dealt to 2008 people seated facing inwards in a circle ...
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Quick View Oundle Selection Test 2. 4 hours 30 minutes. 1. Let m be a fixed integer greater than 1. The sequence 51:0, 931,172, . . . is. defined as follows: Find the greatest ...
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Quick View FST1 2009. OCR. TCC April 4. 1. Find all prime numbers p such that. 2p_1 — 1. P is a perfect square. 2. Each point of the plane is painted one of three colours.
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Quick View Next Selection Test: Exam 4. IMO camp, Oundle School g " 28- -2008 W. M i" i <; (»1*””/1:5 eke'. \jPr0blem 1O Consider an infinite sequence of distinct positive ...
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Quick View Next Selection Test: Paper 4. Oundle School, Northamptonshire. 6th June 2012. 1. Determine all pairs (f,g) of functions from the set of real numbers to itself that ...
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Quick View Oundle Test 4. 30 May 2007. 1. Consider triangle ABC'. B1 is on the line AC and the line BB1 passes through the incentre I. The point C1 is similarly defined.
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Quick View UK IMO Next Selection Test 1. Oundle 2006. 1. Does there exist a bounded function f : R → R with f(1) > 0 satisfying f(x + y)2 ≥ f(x)2 + f(2xy) + f(y)2 for all x, y ...
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Quick View 31 May 2010 – NST 2. 31 May 2010. 1. The sequence (ai) is defined by a0 = 2, a1 = 1, and for n ≥ 2 we let an = an−1 + an−2. Show that if p is a prime divisor ...
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Quick View First Selection Test: Exam 2. IMO camp, Trinity College Cambridge. 2-iv-2007. Problem 1 Let (1, b be positive integers such that for every positive integer ...
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Quick View Next Selection Test: 4 hours 30 minutes. Oundle, May 27, 2003 . Let p1,p2, . . . , pn be distinct prime numbers greater than 3. Show that. 2P””"4”' + 1 has at least ...
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Quick View FSTl 2006. Trinity College Cambridge. 1. Let E be the intersection of the diagonals of the cyclic quadrilateral. ABCD. Let F and G denote the respective ...
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Quick View FST2 2009. OCR. TCC April 6. 1. Triangle ABC has a right-angle at C, and the point M' on AB is strictly between A and B. Let S, S1 and S2 denote the ...
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Quick View 2 Dec 2010 – UKMT, School of Mathematics Satellite, University of Leeds, Leeds LS2 9JT. Tel: 0113 343 2339 Fax: 0113 343 5500 Email: ...
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Quick View UK IMO FST1. Trinity College, Cambridge. 9 April 2005. 1. An infinite sequence a0,a1,a2,... of real numbers satisfies the condition an = |an+1 − an+2| for every n ...
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Quick View NST 1, Eton and Oundle, 2009. 1. The convex quadrilateral ABCD has the property that lAB{ : |AC] =1. |BD{. Let P be the intersection point of its diagonals.
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Quick View European Girls' Mathematical Olympiad. EGMO | 2012. UK Mathematical Olympiad for Girls. 23 June 2011. Instructions. Supported by. • Time allowed: 3 hours.
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Quick View United Kingdom Mathematics Trust. UK Mathematical Olympiad for Girls. 20 September 2012. Instructions. • Time allowed: 3 hours. • Full written solutions – not ...
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Quick View 16 Jul 2012 – UNITED KINGDOM MATHEMATICS TRUST. The UKMT is a company limited by guarantee and registered in England and Wales. Registered ...
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Quick View THE 4th ROMANIAN MASTER OF MATHEMATICS COMPETITION. DAY 2: SATURDAY, FEBRUARY 26, 2011, BUCHAREST. Language: English. Problem 4.
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