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: 8

2018 Romanian Masters in Mathematics, 1
Tags: geometry , RMM , RMM 2018 , auyesl
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

2018 Romanian Master of Mathematics, 2
Tags: algebra , polynomial , RMM , RMM 2018
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying $$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$

2018 Romanian Master of Mathematics, 6
Tags: geometry , RMM 2018
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.

2018 Romanian Masters in Mathematics, 2
Tags: algebra , polynomial , RMM , RMM 2018
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying $$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$

2018 Romanian Master of Mathematics, 3
Tags: RMM , RMM 2018 , combinatorics
Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

2018 Romanian Master of Mathematics, 1
Tags: geometry , RMM , RMM 2018 , auyesl
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

2018 Romanian Masters in Mathematics, 6
Tags: geometry , RMM 2018
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.

2018 Romanian Masters in Mathematics, 3
Tags: RMM , RMM 2018 , combinatorics
Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?