MathDB

2

Part of 2014 Turkey Team Selection Test

Problems(3)

Hard Functional Equation

Source: Turkey TST 2014 Day 1 Problem 2

3/12/2014
Find all ff functions from real numbers to itself such that for all real numbers x,yx,y the equation f(f(y)+x2+1)+2x=y+(f(x+1))2f(f(y)+x^2+1)+2x=y+(f(x+1))^2 holds.
functionsymmetryalgebra proposedalgebrafunctional equation
Concurency

Source: Turkey TST 2014 Day 2 Problem 5

3/12/2014
A circle ω\omega cuts the sides BC,CA,ABBC,CA,AB of the triangle ABCABC at A1A_1 and A2A_2; B1B_1 and B2B_2; C1C_1 and C2C_2, respectively. Let PP be the center of ω\omega. A′A' is the circumcenter of the triangle A1A2PA_1A_2P, B′B' is the circumcenter of the triangle B1B2PB_1B_2P, C′C' is the circumcenter of the triangle C1C2PC_1C_2P. Prove that AA′,BB′AA', BB' and CC′CC' concur.
geometrycircumcirclegeometric transformationhomothetytrigonometryconicsinequalities
Natural Number Sequence

Source: Turkey TST 2014 Day 3 Problem 8

3/12/2014
a1=−5a_1=-5, a2=−6a_2=-6 and for all n≥2n \geq 2 the (an)∞n=1{(a_n)^\infty}_{n=1} sequence defined as, an+1=an+(a1+1)(2a2+1)(3a3+1)⋯((n−1)an−1+1)((n2+n)an+2n+1)).a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)). If a prime pp divides nan+1na_n+1 for a natural number n, prove that there is a integer mm such that m2≡5(modp)m^2\equiv5(modp)
number theory proposednumber theory