This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 9

2015 IFYM, Sozopol, 5
Tags: number theory , Taiwan , Taiwan TST 2013
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.

2013 Taiwan TST Round 1, 3
Tags: functional equation , algebra , Taiwan , Taiwan TST 2013
Find all $g:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in R$, \[(4x+g(x)^2)g(y)=4g(\frac{y}{2}g(x))+4xyg(x)\]

2013 Taiwan TST Round 1, 2
Tags: Inequality , algebra , Taiwan , Taiwan TST 2013
Prove that for positive reals $a,b,c$, \[\frac{8a^2+2ab}{(b+\sqrt{6ac}+3c)^2}+\frac{2b^2+3bc}{(3c+\sqrt{2ab}+2a)^2}+\frac{18c^2+6ac}{(2a+\sqrt{3bc}+b})^2\geq 1\]

2013 Taiwan TST Round 1, 1
Tags: geometry , Taiwan , Taiwan TST 2013
Let P be a point in an acute triangle $ABC$, and $d_A, d_B, d_C$ be the distance from P to vertices of the triangle respectively. If the distance from P to the three edges are $d_1, d_2, d_3$ respectively, prove that \[d_A+d_B+d_C\geq 2(d_1+d_2+d_3)\]

2013 Taiwan TST Round 1, 1
Tags: geometry , circumcircle , rhombus , Taiwan , Taiwan TST 2013
Let $\Delta ABC$ be a triangle with $AB=AC$ and $\angle A = \alpha$, and let $O,H$ be its circumcenter and orthocenter, respectively. If $P,Q$ are points on $AB$ and $AC$, respectively, such that $APHQ$ forms a rhombus, determine $\angle POQ$ in terms of $\alpha$.

2013 Taiwan TST Round 1, 2
Tags: polyomino , combinatorics , Taiwan , Taiwan TST 2013
A V-tromino is a diagram formed by three unit squares.(As attachment.) (a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes? (b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?

2013 Taiwan TST Round 1, 2
Tags: number theory , Taiwan , Taiwan TST 2013
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.

2013 Taiwan TST Round 1, 1
Tags: Taiwan , Taiwan TST 2013
Is it possible to divide $\mathbb{N}$ into six disjoint sets $A_1, A_2, A_3, A_4, A_5, A_6$, such that $x,y,z$ are not in the same set if $x+2y=5z$?

2013 Taiwan TST Round 1, 1
Tags: number theory , prime , prime numbers , Taiwan , Taiwan TST 2013
Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?