MathDB

6

Part of 2016 China Team Selection Test

Problems(3)

Counting Subsets that sum to zero mod m

Source: China 2016 TST Day 2 Q6

3/16/2016
Let m,nm,n be naturals satisfying n≥m≥2n \geq m \geq 2 and let SS be a set consisting of nn naturals. Prove that SS has at least 2n−m+12^{n-m+1} distinct subsets, each whose sum is divisible by mm. (The zero set counts as a subset).
Combinatorial Number Theorynumber theorycombinatorics
Perpendicular following tangent circles

Source: China Team Selection Test 2016 Test 2 Day 2 Q6

3/21/2016
The diagonals of a cyclic quadrilateral ABCDABCD intersect at PP, and there exist a circle Γ\Gamma tangent to the extensions of AB,BC,AD,DCAB,BC,AD,DC at X,Y,Z,TX,Y,Z,T respectively. Circle Ω\Omega passes through points A,BA,B, and is externally tangent to circle Γ\Gamma at SS. Prove that SP⊥STSP\perp ST.
geometrycyclic quadrilateralharmonic division
Function preserving triangle inequality

Source: China Team Selection Test 2016 Test 3 Day 2 Q6

3/26/2016
Find all functions f:R+→R+f: \mathbb R^+ \rightarrow \mathbb R^+ satisfying the following condition: for any three distinct real numbers a,b,ca,b,c, a triangle can be formed with side lengths a,b,ca,b,c, if and only if a triangle can be formed with side lengths f(a),f(b),f(c)f(a),f(b),f(c).
algebrafunction