Problems

Number Theory

  1. Let P,Q,R be primes such that P,Q,R,P+Q,P-Q are all two digit numbers, which use the digits 0 to 9 exactly once. What is R?
  2. Let a,b,c,d be positive integers satisfying ad=bc. Prove that a+b+c+d is composite.
  3. 10 consecutive integers are given. Prove that there is one that is coprime to the others
  4. There are 100 open lockers labelled 1 to 100, corresponding to people also labelled 1 to 100. Person n can either change the state of every nth locker, or abstain. In the end, only locker 1 is closed. How many people abstained?

Combinatorics

  1. Find 16 points, such that they make 15 quadruplets of collinear points.
  2. If n circles are on the plane, at most how many regions can we create?
  3. In a 6 by 6 grid, what is the largest number of squares that can be colored black such that no four black squares form the corners of a rectangle with sides parallel to the grid?
  4. How many queens can be placed on an n by n chessboard such that each queen attacks exactly two other queens?

Algebra

  1. The first four terms of an arithmetic sequence are x+y, x-y, xy, and x/y in that order. What is the fifth term?
  2. Let a,b,c be non-zero real numbers such that a²-b²=bc and b²-c²=ac. Prove that a²-c²=ab.
  3. Prove that any function can be written as the sum of an odd function and an even function.
  4. Let a,b>0. If Q,A,G,H are the quadratic, arithmetic, geometric, and harmonic mean of a and b respectively, prove that Q+H≥A+G.

Geometry

  1. Let the incircle of right triangle ABC (B=90°) be tangent to AC at D. Prove that the area of ABC is AD*DC.
  2. ABCD circumscribes an ellipse with focus F. Prove that ∠AFB+∠CFD=180°.
  3. Given a 19° angle, construct a 1° angle using ruler and compass.
  4. Let ABC and DEF be similar and equally oriented triangles. If E lies on AB and F lies on AC, prove that the loci of D is a line.
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