Mock Usajmo Shortlist
Mock Usajmo Shortlist
Mock Usajmo Shortlist
1 Algebra
Problem 1. Let x, y, z be positive real numbers. Prove that (x2 + y 2 + z 2 )2 ≥ 3xyz(x + y + z).
Problem 2. Find all functions f : R → R satisfying
2 Geometry
Problem 5. Four points A, B, C, D lie on a circle with center O. Points X and Y are chosen
on AB and AD respectively so that CX⊥CD and CY ⊥BC. Prove that points O, X, Y are
collinear.
Problem 6. Let ABC be a triangle, and X, Y be points on the circumcircle of ABC such that
AX = AY . Let B 0 and C 0 be the intersections of XY with AB and AC, respectively. Let P
be an arbitrary point in the plane, and let Q be the second intersection of the circumcircles of
BB 0 P and CC 0 P . Prove that P Q passes through A.
3 Combinatorics
Problem 7. Take any arbitrary set A0 ⊂ N, and recursively define Aj+1 ⊆ N for each j ≥ 0
as the set containing the positive integer k if and only if
1
is odd. For each positive integer M , prove that there are infinitely many positive integers n so
that all elements of A0 that are less than M are also elements of An .
4 Number Theory
Problem 9. Does there exist a positive integer m such that there exists positive itnegers n1 , n2
and n3 such that the first 4 digits of mn1 , mn2 , mn3 are 1520, 2015 and 5201?
Problem 10. A sequence of numbers a1 , a2 , . . . has the following property: if the last digit of
10k − 1 an − k
an is k, an+1 = + . Given that a1 = 20155102, show that all numbers in this
9 10
sequence have at least 5 digits and find a2015 .