Found problems: 85335
2007 IMC, 3
Let $ C$ be a nonempty closed bounded subset of the real line and $ f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $ p\in C$ such that $ f(p) \equal{} p$.
(A set is closed if its complement is a union of open intervals. A function $ g$ is nondecreasing if $ g(x)\le g(y)$ for all $ x\le y$.)
2010 ISI B.Stat Entrance Exam, 1
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
2023 VN Math Olympiad For High School Students, Problem 4
Determine whether or not the length of symmedian is not greater than the length of the angle bisector drawn from the same vertex?
2015 British Mathematical Olympiad Round 1, 4
James has a red jar, a blue jar and a pile of $100$ pebbles. Initially both jars are empty. A move consists of moving a pebble from the pile into one of the jars or returning a pebble from one of the jars to the pile. The numbers of pebbles in the red and blue jars determine the state of the game. The followwing conditions must be satisfied:
(a) The red jar may never contain fewer pebbles than the blue jar;
(b) The game may never be returned to a previous state.
What is the maximum number of moves that James can make?
2014 Online Math Open Problems, 30
Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$.
[i]Proposed by Yang Liu[/i]
2014 VTRMC, Problem 4
Suppose we are given a $19\times19$ chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^2-1=360$ squares with $4\times1$ and $1\times4$ rectangles? (So that each of the $360$ squares is covered by exactly one rectangle.) Justify your answer.
2024 HMNT, 21
Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino.
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2003 Paraguay Mathematical Olympiad, 3
Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in $33$ years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.
2022 HMNT, 7
Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation
$$gcd(a, b) \cdot a + b^2 = 10000.$$
2021 Iran Team Selection Test, 5
Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$.
Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$.
Proposed by [i]Alireza Dadgarnia[/i]
2016 Azerbaijan Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2010 Dutch BxMO TST, 4
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
Russian TST 2017, P1
A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city
2017 Dutch IMO TST, 4
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$
for all $x, y \in \mathbb{R}$.
1982 Vietnam National Olympiad, 1
Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.
2022 IMAR Test, 1
Find all pairs of primes $p, q<2023$ such that $p \mid q^2+8$ and $q \mid p^2+8$.
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
2013 Vietnam National Olympiad, 3
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.
[b]a)[/b] Prove that $D,I,J$ collinear.
[b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.
Kvant 2020, M1000
A polyline $AMB$ is inscribed in the arc $AB{}$, consisting of two segments, and $AM>MB$. Let $K$ be the midpoint of the arc $AB{}$. Prove that the foot $H{}$ of the perpendicular from $K$ onto $AM$ divides the polyline in two equal segments: \[AH=HM+MB.\][i]Discovered by Archimedes[/i]
LMT Guts Rounds, 33
Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$
2018 Kyiv Mathematical Festival, 4
Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?
2016 Korea Winter Program Practice Test, 3
$p, q, r$ are natural numbers greater than 1.
There are $pq$ balls placed on a circle, and one number among $0, 1, 2, \cdots , pr-1$ is written on each ball, satisfying following conditions.
(1) If $i$ and $j$ is written on two adjacent balls, $|i-j|=1$ or $|i-j|=pr-1$.
(2) $i$ is written on a ball $A$. If we skip $q-1$ balls clockwise from $A$ and see $q^{th}$ ball, $i+r$ or $i-(p-1)r$ is written on it. (This condition is satisfied for every ball.)
If $p$ is even, prove that the number of pairs of two adjacent balls with $1$ and $2$ written on it is odd.
2009 AMC 8, 24
The letters $ A$, $ B$, $ C$ and $ D$ represent digits. If $ \begin{tabular}{ccc} &A&B \\ \plus{}&C&A \\ \hline &D&A \end{tabular}$ and $ \begin{tabular}{ccc} &A&B \\ \minus{}&C&A \\ \hline &&A \end{tabular}$, what digit does $ D$ represent?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
2019 Novosibirsk Oral Olympiad in Geometry, 3
Equal line segments are marked in triangle $ABC$. Find its angles.
[img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]
2024 Azerbaijan BMO TST, 5
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
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[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
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Prove that $\max(a_1,a_{2023})\ge 507$.