Olympiad problems

2002 AMC 10, 22

Tags:

Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$. $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

2023 Moldova EGMO TST, 6

Tags: geometry

Let there be a square $ABCD$. Points $E$ and $F$ are on sides $(BC)$ and $(AB)$ such that $BF=CE$. LInes $AE$ and $CF$ intersect in point $G$. Prove that $EF$ and $DG$ are perpendicular.

2012 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: inequalities , algebra

Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$ where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers

KoMaL A Problems 2024/2025, A. 884

Tags: combinatorics , numbers in a table , graph theory

We fill in an $n\times n$ table with real numbers such that the sum of the numbers in each row and each coloumn equals $1$. For which values of $K$ is the following statement true: if the sum of the absolute values of the negative entries in the table is at most $K$, then it's always possible to choose $n$ positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries. [i]Proposed by Dávid Bencsik, Budapest[/i]

2020 SMO, 5

Tags: geometry

In triangle $\triangle ABC$, let $E$ and $F$ be points on sides $AC$ and $AB$, respectively, such that $BFEC$ is cyclic. Let lines $BE$ and $CF$ intersect at point $P$, and $M$ and $N$ be the midpoints of $\overline{BF}$ and $\overline{CE}$, respectively. If $U$ is the foot of the perpendicular from $P$ to $BC$, and the circumcircles of triangles $\triangle BMU$ and $\triangle CNU$ intersect at second point $V$ different from $U$, prove that $A, P,$ and $V$ are collinear. [i]Proposed by Andrew Wen and William Yue[/i]

2014 Korea National Olympiad, 2

Tags: function , algebra , functional equation

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

1956 Putnam, B3

Tags: geometry , 3d geometry , sphere , tetrahedron

A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.

2007 Today's Calculation Of Integral, 182

Tags: calculus , integration , geometry , function , domain , inequalities , algebra

Find the area of the domain of the system of inequality \[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]

III Soros Olympiad 1996 - 97 (Russia), 10.6

Tags: geometry , rhombus

There is a rhombus with acute angle $b$ and side $a$. Two parallel lines, the distance between which is equal to the height of the rhombus, intersect all four sides of the rhombus. What can be the sum of the perimeters of two triangles cut off from a rhombus by straight lines? (These two triangles lie outside the strip between parallel lines.)

2019 AMC 10, 25

Tags: factorial

For how many integers $n$ between $1$ and $50$, inclusive, is \[ \frac{(n^2-1)!}{(n!)^n} \]an integer? (Recall that $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$