MathDB

1

Part of 2015 Iran Team Selection Test

Problems(3)

Easy polynomial

Source: Iran TST 2015, first exam, day 1 problem 1

5/17/2015
Find all polynomials P,Q∈Q[x]P,Q\in \Bbb{Q}\left [ x \right ] such that P(x)3+Q(x)3=x12+1.P(x)^3+Q(x)^3=x^{12}+1.
algebrapolynomialalgebra proposed
4 Variable Inequality

Source: Iran TST 2015 - second exam - first day problem 1

6/12/2015
a,b,c,da,b,c,d are positive numbers such that ∑cyc1ab=1\sum_{cyc} \frac{1}{ab} =1. Prove that : abcd+16≥8(a+c)(1a+1c)+8(b+d)(1b+1d)abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}
inequalities unsolvedinequalities
easy geometry

Source: Iran team selection test third exam p1

6/4/2015
Point AA is outside of a given circle ω\omega. Let the tangents from AA to ω\omega meet ω\omega at S,TS, T points X,YX, Y are midpoints of AT,ASAT, AS let the tangent from XX to ω\omega meet ω\omega at R≠TR\neq T. points P,QP, Q are midpoints of XT,XRXT, XR let XY∩PQ=K,SX∩TK=LXY\cap PQ=K, SX\cap TK=L prove that quadrilateral KRLQKRLQ is cyclic.
geometryradical axis