Found problems: 85335
2014 Turkey EGMO TST, 6
For a given integer $n\ge3$, let $S_1, S_2,\ldots,S_m$ be distinct three-element subsets of the set $\{1,2,\ldots,n\}$ such that for each $1\le i,j\le m; i\neq j$ the sets $S_i\cap S_j$ contain exactly one element. Determine the maximal possible value of $m$ for each $n$.
2020 Cono Sur Olympiad, 3
Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.
2011 Junior Balkan Team Selection Tests - Moldova, 4
In the Cartesian $xOy$ coordinate system the points $A (36, 0)$, $A_1 (10, 0)$ are given, $B (0, 36)$, $B_1 (0, 10)$, $C (-36, 0)$, $C_1 (-10, 0)$, $D (0, -36)$, $D_1 (0, -10)$. A point of the plane is called [i]lattice[/i] if it has integer coordinates. Determine the number of lattice points that are located inside the square $ABCD$, but outside the square $A_1B_1C_1D_1$
2014 PUMaC Number Theory B, 5
Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 -1$ for some positive integer $n$.
1977 IMO Shortlist, 4
Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.
2022 Harvard-MIT Mathematics Tournament, 3
Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly $10$ operations.
JBMO Geometry Collection, 2001
Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$.
[i]Bulgaria[/i]
2000 Mongolian Mathematical Olympiad, Problem 4
In a country with $n$ towns, the distance between the towns numbered $i$ and $j$ is denoted by $x_{ij}$. Suppose that the total length of every cyclic route which passes through every town exactly once is the same. Prove that there exist numbers $a_i,b_i$ ($i=1,\ldots,n$) such that $x_{ij}=a_i+b_j$ for all distinct $i,j$.
1957 AMC 12/AHSME, 27
The sum of the reciprocals of the roots of the equation $ x^2 \plus{} px \plus{} q \equal{} 0$ is:
$ \textbf{(A)}\ \minus{}\frac{p}{q} \qquad
\textbf{(B)}\ \frac{q}{p}\qquad
\textbf{(C)}\ \frac{p}{q}\qquad
\textbf{(D)}\ \minus{}\frac{q}{p}\qquad
\textbf{(E)}\ pq$
2022 IOQM India, 1
Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$ and $C$ lies on $L_2$. Find the area of the square.
2010 Saudi Arabia IMO TST, 1
Let $ABC$ be a triangle with $\angle B \ge 2\angle C$. Denote by $D$ the foot of the altitude from $A$ and by $M$ be the midpoint of $BC$. Prove that $DM \ge \frac{AB}{2}$.
2025 Euler Olympiad, Round 2, 6
For any subset $S \subseteq \mathbb{Z}^+$, a function $f : S \to S$ is called [i]interesting[/i] if the following two conditions hold:
[b]1.[/b] There is no element $a \in S$ such that $f(a) = a$.
[b]2.[/b] For every $a \in S$, we have $f^{f(a) + 1}(a) = a$ (where $f^{k}$ denotes the $k$-th iteration of $f$).
Prove that:
[b]a) [/b]There exist infinitely many interesting functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$.
[b]b) [/b]There exist infinitely many positive integers $n$ for which there is no interesting function
$$
f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}.
$$
[i]Proposed by Giorgi Kekenadze, Georgia[/i]
KoMaL A Problems 2017/2018, A. 720
We call a positive integer [i]lively[/i] if it has a prime divisor greater than $10^{10^{100}}$. Prove that if $S$ is an infinite set of lively positive integers, then it has an infinite subset $T$ with the property that the sum of the elements in any finite nonempty subset of $T$ is a lively number.
1996 Romania National Olympiad, 1
Find all pairs of real numbers $(x, y) $ such that:
a) $x\ge y\ge1$
b) $2x^2-xy-5x +y + 4 = 0 $
2018 Pan African, 6
A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?
2017 ASDAN Math Tournament, 23
Ben creates an $8\times8$ grid of coins, where each coin faces heads with probability $\tfrac{1}{2}$, and tails with probability $\tfrac{1}{2}$. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?
2014 Paraguay Mathematical Olympiad, 1
Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?
1983 AMC 12/AHSME, 3
Three primes $p,q,$ and $r$ satisfy $p+q = r$ and $1 < p < q$. Then $p$ equals
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 17 $
2009 Serbia Team Selection Test, 3
Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.
2021 China Girls Math Olympiad, 5
Proof that if $4$ numbers (not necessarily distinct) are picked from $\{1, 2, \cdots, 20\}$, one can pick $3$ numbers among them and can label these $3$ as $a, b, c$ such that $ax \equiv b \;(\bmod\; c)$ has integral solutions.
2002 France Team Selection Test, 1
There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.
2018 MOAA, 3
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
2005 AMC 12/AHSME, 12
A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$
2021-2022 OMMC, 4
If $x, y, z$ satisfy $x+y+z = 12, \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$ and $x^3+y^3+z^3 = -480,$ find $$x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2.$$
[i]Proposed by Mahith Gottipati[/i]
1993 Iran MO (2nd round), 3
Let $f(x)$ and $g(x)$ be two polynomials with real coefficients such that for infinitely many rational values of $x$, the fraction $\frac{f(x)}{g(x)}$ is rational. Prove that $\frac{f(x)}{g(x)}$ can be written as the ratio of two polynomials with rational coefficients.