Apr 7, 2011

Singapore Mathematical Olympiad questions


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    Singapore International Mathematical Olympiad National Team ...
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    Singapore International Mathematical Olympiad. National Team Selection Tests 1994/95 - 2003/2004. 1994/95. 1.1.* Let N = {1,2,3, . ...
    sms.math.nus.edu.sg/Simo/199567890123.pdf
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    Singapore International Mathematical Olympiad Committee 22-9-2001 ...
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    Singapore International Mathematical Olympiad Committee. 22-9-2001. 1. Consider the set of integers A = {2a3b5c : 0 ≤ a, b, c ≤ 5}. ...
    sms.math.nus.edu.sg/Simo/training2001.pdf
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    Singapore International Mathematical Olympiad National Team ...
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    Singapore International Mathematical Olympiad. National Team Selection Test I 2006. Time allowed: 4.5 hours. 12 May 2006. No calculator is allowed ...
    sms.math.nus.edu.sg/Simo/selection_test_2006.pdf
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    Competition corner
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    Competition corner. All communications about this column should be sent to Dr. Tay Tiong. Seng, Department of Mathematics, National University of Singapore, ...
    sms.math.nus.edu.sg/simo/med022.pdf
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    Training problems 10 April 2003 14. Let A be a set with 8 elements ...
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    Training problems 10 April 2003. 14. Let A be a set with 8 elements. Find the maximum number of distinct 3-element subsets of A such that the intersection ...
    sms.math.nus.edu.sg/Simo/training2003/trainging_3q.pdf
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    1 April - Training problems 1 April 2003 8. In a group of ...
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    Training problems 1 April 2003. 8. In a group of interpreters each one speaks one or several languages, 24 of them speak ...
    sms.math.nus.edu.sg/Simo/training2003/training_2q.pdf
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    Singapore International Mathematical Olympiad 2009 Senior Team ...

     - 4:35pm
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    Singapore International Mathematical Olympiad 2009. Senior Team Training. Take Home Test Solutions. 1. For any positive integer k, let f(k) be the number of ...
    sms.math.nus.edu.sg/Simo/2009/2009SeniorTestSoln.pdf
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    200e +FilE#fi+Fffi[Ufrffi#-
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    200e +FilE#fi+Fffi[Ufrffi#- ffi-x r. ft u E_--+ *r:*WRn *-f$€'FE4'lLREt+#tJffiF-A. ffqE, xf{tffiiE. #W. n, #$#f:. n lY*fr.*,.flx), ]tfr1F- flx)fr\FfrH 6W. ...
    sms.math.nus.edu.sg/Simo/CWMO/cwmo2009_questions.pdf
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    44th International Mathematical Olympiad
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    44th International Mathematical Olympiad. Tokyo, Japan, July 2003. 1. Let A be a 101-element subset of the set S = {1,2,3,...,1000000}. Prove that there ...
    sms.math.nus.edu.sg/Simo/IMO_Problems/03.pdf
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    Singapore International Mathematical Olympiad 2009 Senior Team ...
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    Singapore International Mathematical Olympiad 2009. Senior Team Training. Quiz. 1. Let p and q be distinct odd primes. Prove that. ∑. 0<j<p/2 ...
    sms.math.nus.edu.sg/Simo/2009/2009SeniorQuizSoln.pdf
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    45th International Mathematical Olympiad

     - 4:22pm
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    45th International Mathematical Olympiad. Athens, Greece, July 2004. 1. Let ABC be an acute-angled triangle with AB = AC. The circle with diameter BC ...
    sms.math.nus.edu.sg/Simo/IMO_Problems/04.pdf
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    Selection Procedure of IMO participants
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    Selection Procedure of IMO participants. SIMO Committee. SIMO Team to participate in IMO. Participate in. SMO(Junior). Junior Training Team. Final Selection ...
    sms.math.nus.edu.sg/Simo/Selection_Flowchart.pdf
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    Singapore International Mathematical Olympiad 2003 National Team ...
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    Singapore International Mathematical Olympiad 2003. National Team Training. Geometry. (8) Let P and Q be the feet of the perpendiculars from the orthocenter ...
    sms.math.nus.edu.sg/Simo/training2003/training20_03_2003.pdf
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    41st International Mathematical Olympiad
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    41st International Mathematical Olympiad. Taejon, Korea, July 2000. 1. Two circles Γ1 and Γ2 intersect at M and N. Let l be the common tangent to Γ1 and Γ2 ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C00.pdf
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    2002 - 43rd International Mathematical Olympiad
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    43rd International Mathematical Olympiad. Glasgow, United Kingdom, July 2002. 1. Let n be a positive integer. let T be the set of points (x, y) in the plane ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C02.pdf
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    40th International Mathematical Olympiad
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    40th International Mathematical Olympiad. Bucharest, Romania, July 1999. 1. Determine all finite sets S of at least three points in the plane which satisfy ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C99.pdf
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    38th International Mathematical Olympiad
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    38th International Mathematical Olympiad. Argentina, July 1997. 1. In the plane the points with integer coordinates are the vertices of unit squares. The ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C97.pdf
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    37th International Mathematical Olympiad
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    37th International Mathematical Olympiad. India, July 1996. 1. Let ABCD be a rectangular board with |AB| = 20, |BC| = 12. The board is divided ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C96.pdf
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    42nd International Mathematical Olympiad

     - 4:22pm
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    42nd International Mathematical Olympiad. Washington DC, United States of America, July 2001. 1. Let ABC be an acute-angled triangle with circumcentre O. ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C01.pdf
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    35th International Mathematical Olympiad
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    35th International Mathematical Olympiad. Hong Kong, July 1994. 1. Let m and n be positive integers. let a1, a2, ..., am be distinct elements of {1,2,...,n} ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C94.pdf
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    39th International Mathematical Olympiad

     - 4:22pm
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    39th International Mathematical Olympiad. Taiwan, July 1998. 1. In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C98.pdf - Similar
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    33th International Mathematical Olympiad
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    33th International Mathematical Olympiad. Russia, July 1992. 1. Find all integers a, b, c with 1 <a<b<c such that (a − 1)(b − 1(c − 1) is a divisor of ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C92.pdf
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    36th International Mathematical Olympiad
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    36th International Mathematical Olympiad. Canada, July 1995. 1. Let A, B, C, D be distinct points on a line, in that order. The circles with diameters ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C95.pdf
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    34th International Mathematical Olympiad
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    34th International Mathematical Olympiad. Turkey, July 1993. 1. Let f(x) = xn + 5xn−1 + 3 where n > 1 is an integer. Prove that f(x) cannot be ...
    sms.math.nus.edu.sg/Simo/IMO_Problems%5C93.pdf - Similar
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    12 July 2006 Problem 1. Let ABC be a triangle with incentre I. A ...
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    12 Jul 2006 ... 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. ...
    sms.math.nus.edu.sg/SIMO/IMO_Participation/imo47/day1.pdf - Similar
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    International Competitions IMO 2007
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    Ha Noi, Vietnam. Day 2 - 26 July 2007. 4 In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the per- ...
    sms.math.nus.edu.sg/SIMO/IMO_Participation/imo48/day2.pdf
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    46th International Mathematical Olympiad
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    46th International Mathematical Olympiad. Second Day. Merida, Mexico, Thursday 14 July 2005. Language: English. Problem 4. Consider the sequence a1,a2,... ...
    sms.math.nus.edu.sg/SIMO/IMO_Participation/imo46/day2.pdf - Similar
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    13 July 2006 Problem 4. Determine all pairs (x, y) of integers ...
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    13 July 2006. Problem 4. Determine all pairs (x, y) of integers such that. 1+2x + 22x+1 = y2. Problem 5. Let P(x) be a polynomial of degree n > 1 with ...
    sms.math.nus.edu.sg/SIMO/IMO_Participation/imo47/day2.pdf - Similar
  29. [PDF] 

    46th International Mathematical Olympiad
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    46th International Mathematical Olympiad. First Day. Merida, Mexico, Wednesday 13 July 2005. Language: English. Problem 1. Six points are chosen on the ...
    sms.math.nus.edu.sg/SIMO/IMO_Participation/imo46/day1.pdf - Similar
  30. [PDF] 

    Singapore International Mathematical Olympiad Training Problems
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    Singapore International Mathematical Olympiad. Training Problems. 18 January 2003. 1. Let M be a point on the segment AB. Squares AMCD and MBEF are erected ...
    sms.math.nus.edu.sg/Simo/training2003/training18_01_2003.pdf
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    Singapore International Mathematical Olympiad Training Problems

     - 2 visits - 4:35pm
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    Singapore International Mathematical Olympiad. Training Problems. 25 January 2003. 1. In a quadrilateral ACGE, H is the intersection of AG and CE, ...
    sms.math.nus.edu.sg/Simo/training2003/training25_01_2003.pdf
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    Training problems 27 March 2003 1. Find all integer solutions in x ...
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    27 Mar 2003 ... Training problems 27 March 2003. 1. Find all integer solutions in x and y of the equation x3 + 27xy + 2009 = y3. Solution. Let y = x + a. ...
    sms.math.nus.edu.sg/Simo/training2003/training27_03_2003.pdf
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    Singapore International Mathematical Olympiad Training Problems
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    Singapore International Mathematical Olympiad. Training Problems. 08 March 2003. 1. Let p be a prime number congruent to 1 modulo 4. Define the set ...
    sms.math.nus.edu.sg/Simo/training2003/training08_03_2003.pdf
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    Singapore International Mathematical Olympiad Training Problems
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    Singapore International Mathematical Olympiad. Training Problems. 15 February 2003. 1. Let n be an odd integer which is not a multiple of 5. ...
    sms.math.nus.edu.sg/Simo/training2003/training15_02_2003.pdf
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    Singapore International Mathematical Olympiad Training Problems

     - 4:33pm
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    Singapore International Mathematical Olympiad. Training Problems. 8 February 2003. 1. Determine whether there exist an integer polynomial f(x) together with ...
    sms.math.nus.edu.sg/Simo/training2003/training08_02_2003.pdf
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    International Mathematical Olympiad

     - 4:33pm
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    Thursday, July 16, 2009. Problem 4. Let ABC be a triangle with AB = AC. The angle bisectors of CAB and ABC meet the sides BC and CA at D and E, respectively ...
    sms.math.nus.edu.sg/Simo/IMO_Participation/imo50/Day2.pdf
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    Language: English Day: 2 51st INTERNATIONAL MATHEMATICAL OLYMPIAD ...
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    Language: English. Day: 2. 51st INTERNATIONAL MATHEMATICAL OLYMPIAD. ASTANA (KAZAKHSTAN), JULY 2-14, 2010. Thursday, July 8, 2010. Problem 4. ...
    sms.math.nus.edu.sg/Simo/IMO_Participation/imo51/Day_2.pdf
  38. [PDF] 

    Language: English Day: 1 51st INTERNATIONAL MATHEMATICAL OLYMPIAD ...
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    Language: English. Day: 1. 51st INTERNATIONAL MATHEMATICAL OLYMPIAD. ASTANA (KAZAKHSTAN), JULY 2-14, 2010. Wednesday, July 7, 2010. Problem 1. ...
    sms.math.nus.edu.sg/Simo/IMO_Participation/imo51/Day_1.pdf
  39. [PDF] 

    International Mathematical Olympiad
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    Thursday, July 17, 2008. Problem 4. Find all functions f : (0, ∞) → (0, ∞) (so, f is a function from the positive real ...
    sms.math.nus.edu.sg/Simo/IMO_Participation/imo49/Day2.pdf
  40. [PDF] 

    International Mathematical Olympiad
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    Wednesday, July 16, 2008. Problem 1. An acute-angled triangle ABC has orthocentre H. The circle passing through H with centre the midpoint of BC intersects ...
    sms.math.nus.edu.sg/Simo/IMO_Participation/imo49/Day1.pdf
  41. [PDF] 

    International Mathematical Olympiad

     - 4:33pm
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    Wednesday, July 15, 2009. Problem 1. Let n be a positive integer and let a1,...,ak (k ≥ 2) be distinct integers in the set ...
    sms.math.nus.edu.sg/Simo/IMO_Participation/imo50/Day1.pdf
  42. [PDF] 

    International Competitions IMO 2007
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    Ha Noi, Vietnam. Day 1 - 25 July 2007. 1 Real numbers a1, a2, :::, an are given. For each i, (1 i n), define di = maxfaj j 1 ...
    sms.math.nus.edu.sg/SIMO/IMO_Participation/imo48/day1.pdf
  43. [PDF] 

    Singapore International Mathematical Olympiad Junior Group Test
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    Singapore International Mathematical Olympiad Junior. Group Test. Take Home Test - Number Theory. 13-8-2005. 1. Suppose a, b are integers satisfying 24a2 +1 ...
    sms.math.nus.edu.sg/Simo/Junior_test2005.pdf
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    Singapore International Mathematical Olympiad National Team ...
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    Singapore International Mathematical Olympiad. National Team Selection Test I 2008. Time allowed: 4.5 hours. 26 April 2008. No calculator is allowed ...
    sms.math.nus.edu.sg/simo/selection_test_2008.pdf

  45. Singapore International Mathematical Olympiad Committee (SIMO ...
    Singapore International Mathematical. Olympiad Committee (SIMO). Annual Training Camp. Programme for. Senior Group. 8 – 13 June 2007 ...
    sms.math.nus.edu.sg/simo/smocamp_2007.pdf
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    Graph Theory Problems/Solns
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    Graph Theory Problems/Solns. 1. There are n participants in a meeting. Among any group of 4 participants, there is one who knows the other three members of ...
    sms.math.nus.edu.sg/simo/training2003/smograph.pdf
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    Singapore International Mathematical Olympiad 2008 Senior Team ...

     - 4:25pm
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    Singapore International Mathematical Olympiad 2008. Senior Team Training. Take Home Test Solutions. 1. Show that the equation 15x2 − 7y2 = 9 has no ...
    sms.math.nus.edu.sg/simo/2008/2008SeniorTestSoln.pdf
  48. [PDF] 

    Singapore International Mathematical Olympiad 2005 Senior Team ...

     - 4:25pm
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    Singapore International Mathematical Olympiad 2005. Senior Team Training. Take Home Test Solutions. 1. Determine which of the numbers. 20011111 + 1 ...
    sms.math.nus.edu.sg/simo/training2005/2005-th.pdf
  49. [PDF] 

    Singapore International Mathematical Olympiad 2008 Senior Team ...

     - 4:25pm
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    Singapore International Mathematical Olympiad 2008. Senior Team Training. Take Home Test. Submit written solutions by 12 April 2008. ...
    sms.math.nus.edu.sg/SIMO/2008/2008SeniorTest.pdf

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