Olympiad problems

2021 Olympic Revenge, 3

Tags: geometry , incenter , anime , Euler Line , Spiral Similarity , auyesl

Let $I, C, \omega$ and $\Omega$ be the incenter, circumcenter, incircle and circumcircle, respectively, of the scalene triangle $XYZ$ with $XZ > YZ > XY$. The incircle $\omega$ is tangent to the sides $YZ, XZ$ and $XY$ at the points $D, E$ and $F$. Let $S$ be the point on $\Omega$ such that $XS, CI$ and $YZ$ are concurrent. Let $(XEF) \cap \Omega = R$, $(RSD) \cap (XEF) = U$, $SU \cap CI = N$, $EF \cap YZ = A$, $EF \cap CI = T$ and $XU \cap YZ = O$. Prove that $NARUTO$ is cyclic.

2009 Albania Team Selection Test, 4

Tags: number theory , relatively prime , number theory unsolved , auyesl

Find all the natural numbers $m,n$ such that $1+5 \cdot 2^m=n^2$.

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 .

2024 Nordic, 3

Tags: algebra , Nordic , auyesl

Find all functions $f: \mathbb{R} \to \mathbb{R}$ $f(f(x)f(y)+y)=f(x)y+f(y-x+1)$ For all $x,y \in \mathbb{R}$

2016 Polish MO Finals, 4

Tags: number theory , Divisibility , order of an element , auyesl

Let $k, n$ be odd positve integers greater than $1$. Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$, then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$.

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 .

2020 USA TSTST, 2

Tags: geometry , circumcircle , incircle , Spiral Similarity , geometry solved , USA TSTST , auyesl

Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{EF}$, and let $M$ be the midpoint of $\overline{BC}$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$. [i]Zack Chroman and Daniel Liu[/i]

2012 JBMO TST - Turkey, 1

Tags: geometry , 3D geometry , number theory , least common multiple , number theory proposed , auyesl

Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$

2020 USA EGMO Team Selection Test, 4

Tags: geometry , Angle Chasing , auyesl

Let $ABC$ be a triangle. Distinct points $D$, $E$, $F$ lie on sides $BC$, $AC$, and $AB$, respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$. Let $Q$ denote the reflection of $P$ across $BC$. Show that $Q$ lies on the circumcircle of $\triangle AEF$. [i]Proposed by Ankan Bhattacharya[/i]