Found problems: 85335
2015 ASDAN Math Tournament, 3
Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$.
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2015 HMNT, 5
Kelvin the Frog is trying to hop across a river. The river has $10$ lilypads on it, and he must hop on them in a specific order (the order is unknown to Kelvin). If Kelvin hops to the wrong lilypad at any point, he will be thrown back to the wrong side of the river and will have to start over. Assuming Kelvin is infinitely intelligent, what is the minimum number of hops he will need to guarantee reaching the other side?
2007 Korea - Final Round, 4
Find all pairs $ (p, q)$ of primes such that $ {p}^{p}\plus{}{q}^{q}\plus{}1$ is divisible by $ pq$.
2011 Saudi Arabia BMO TST, 4
Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$, let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$.
Prove that $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$
2002 Turkey MO (2nd round), 3
Graph Airlines $ (GA)$ operates flights between some of the cities of the Republic of Graphia. There are at least three $ GA$ flights from each city, and it is possible to travel from any city in Graphia to any city in Graphia using $ GA$ flights. $ GA$ decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel between any two cities using $ GA$ flights, yet at least $ 2/9$ of the cities have only one flight.
2017 Estonia Team Selection Test, 3
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2005 Austrian-Polish Competition, 8
Given the sets $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$, consider functions $f:R_{mn}\to \{-1,0,1\}$ with the following property: for each quadruple of points $A_1,A_2,A_3,A_4\in R_{mn}$ which form a square with side length $0<s<3$, we have
$$f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.$$
For each pair $(m,n)$ of positive integers, determine $F(m,n)$, the number of such functions $f$ on $R_{mn}$.
II Soros Olympiad 1995 - 96 (Russia), 10.7
Three straight lines $\ell_1$, $\ell_2$ and $\ell_3$, forming a triangle, divide the plane into $7$ parts. Each of the points $M_1$, $M_2$ and $M_3$ lies in one of the angles, vertical to some angle of the triangle. The distance from $M_1$ to straight lines $\ell_1$, $\ell_2$ and $\ell_3$ are equal to $7,3$ and $1$ respectively The distance from $M_2$ to the same lines are $4$, $1$ and $3$ respectively. For $M_3$ these distances are $3$, $5$ and $2$. What is the radius of the circle inscribed in the triangle?
[hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.[/hide]
2017 Taiwan TST Round 1, 1
Find all polynomials $P$ with real coefficients which satisfy
\[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]
2022 Rioplatense Mathematical Olympiad, 3
Let $n$ be a positive integer. Given a sequence of nonnegative real numbers $x_1,\ldots ,x_n$ we define the [i]transformed sequence[/i] $y_1,\ldots ,y_n$ as follows: the number $y_i$ is the greatest possible value of the average of consecutive terms of the sequence that contain $x_i$. For example, the transformed sequence of $2,4,1,4,1$ is $3,4,3,4,5/2$.
Prove that
a) For every positive real number $t$, the number of $y_i$ such that $y_i>t$ is less than or equal to $\frac{2}{t}(x_1+\cdots +x_n)$.
b) The inequality $\frac{y_1+\cdots +y_n}{32n}\leq \sqrt{\frac{x_1^2+\cdots +x_n^2}{32n}}$ holds.
2025 Harvard-MIT Mathematics Tournament, 23
Regular hexagon $ABCDEF$ has side length $2.$ Circle $\omega$ lies inside the hexagon and is tangent to segments $\overline{AB}$ and $\overline{AF}.$ There exist two perpendicular lines tangent to $\omega$ that pass through $C$ and $E,$ respectively. Given that these two lines do not intersect on line $AD,$ compute the radius of $\omega.$
2015 ASDAN Math Tournament, 10
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find
$$\max\{a+b-c,a-b+c,-a+b+c\}.$$
2017 May Olympiad, 4
We consider all $7$-digit numbers that are obtained by swapping in all ways Possible digits of $1234567$. How many of them are divisible by $7$?
2000 Tournament Of Towns, 2
Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$.
(A Spivak)
1999 Mongolian Mathematical Olympiad, Problem 4
Maybe well known:
$p$ a prime number, $n$ an integer. Prove that $n$ divides $\phi(p^n-1)$ where $\phi(x)$ is the Euler function.
2023 Romania EGMO TST, P3
In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .
2010 Contests, 2
Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality
\[
\frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\]
does holds.
2018 Dutch IMO TST, 2
Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.
1964 All Russian Mathematical Olympiad, 048
Find all the natural $n$ such that $n!$ is not divisible by $n^2$.
2008 USAMO, 5
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
2014 Junior Balkan Team Selection Tests - Moldova, 7
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.
2024 Indonesia TST, A
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
2008 Denmark MO - Mohr Contest, 2
If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$Show that $6$ adds up to $pqr$ .
2009 VTRMC, Problem 5
Suppose $A,B\in M_3(\mathbb C)$, $B\ne0$, and $AB=0$. Prove that there exists $D\in M_3(\mathbb C)$ with $D\ne0$ such that $AD=DA=0$.
2014 ELMO Shortlist, 4
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]