Olympiad problems

2017 Romania National Olympiad, 3

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

2023 Middle European Mathematical Olympiad, 2

Tags: inequalities , algebra

If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?

2018 Germany Team Selection Test, 3

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

2023 Malaysia IMONST 2, 2

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Ivan bought $50$ cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number $100000$. How many cats of each breed does he have?

1987 AMC 12/AHSME, 15

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If $(x, y)$ is a solution to the system \[ xy=6 \qquad \text{and} \qquad x^2y+xy^2+x+y=63, \] find $x^2+y^2.$ $ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ \frac{1173}{32} \qquad\textbf{(C)}\ 55 \qquad\textbf{(D)}\ 69 \qquad\textbf{(E)}\ 81 $

2017 ASDAN Math Tournament, 10

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Alice lives on a continent with $6$ countries labeled $1$ through $6$. Each country randomly chooses one other country to allow entry from. Alice can travel to any country that allows entry from the country she is currently in, and can travel along a path through multiple countries in this manner. If Alice starts in county $1$, what is the expected number of countries that she can reach (including country $1$)?

2019 International Zhautykov OIympiad, 2

Tags: algebra , inequalities , number theory

Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

2019 Taiwan APMO Preliminary Test, P6

Tags: algebra , functional equation

Let $\mathbb{N}$ denote the set of all positive integers.Function $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$ satisfies :for any two distinct positive integer $a,b$, we have $$f(a)+f(b)-f(a+b)=2019$$ (1)Find $f(0)$ (2)Let $a_1,a_2,...,a_{100}$ be 100 positive integers (they are pairwise distinct), find $f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})$

2007 Korea Junior Math Olympiad, 8

Tags: prime , positive integer , divisor , number theory

Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infinite number of [i]Primes of the Year[/i].

2016 Postal Coaching, 2

Tags: function , functional equation , algebra

Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$ $$f(xf(y) - yf(x)) = f(xy) - xy.$$