Found problems: 85335
2006 ITAMO, 2
Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.
2020 Thailand Mathematical Olympiad, 9
Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.
2011 IFYM, Sozopol, 6
In a group of $n$ people each one has an Easter Egg. They exchange their eggs in the following way: On each exchange two people exchange the eggs they currently have. Each two exchange eggs between themselves at least once. After a certain amount of such exchanges it turned out that each one of the $n$ people had the same egg he had from the beginning. Determine the least amount of exchanges needed, if:
a) $n=5$;
b) $n=6$.
2020 CMIMC Combinatorics & Computer Science, 1
The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?
2005 IMO Shortlist, 7
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
2010 ELMO Shortlist, 6
For all positive real numbers $a,b,c$, prove that \[\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.\]
[i]In-Sung Na.[/i]
2012 CHMMC Spring, 9
Let $S$ be a square of side length $1$, one of whose vertices is $A$. Let $S^+$ be the square obtained by rotating S clockwise about $A$ by $30^o$ . Let $S^-$ be the square obtained by rotating S counterclockwise about $A$ by $30^o$. Compute the total area that is covered by exactly two of the squares $S$, $S^+$, $S^-$. Express your answer in the form $a + b\sqrt3$ where $a, b$ are rational numbers.
1996 IMO Shortlist, 4
Find all positive integers $ a$ and $ b$ for which
\[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]
2013 AMC 10, 4
When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?
$\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$
1994 China Team Selection Test, 1
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
2006 Thailand Mathematical Olympiad, 9
Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$
2021 Putnam, B4
Let $F_0,F_1,\dots$ be the sequence of Fibonacci numbers, with $F_0=0,F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n \ge 2$. For $m>2$, let $R_m$ be the remainder when the product $\prod_{k=1}^{F_m-1} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.
2016 Brazil Team Selection Test, 5
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$.
[i]Fajar Yuliawan, Indonesia[/i]
1989 Putnam, A4
Is there a gambling game with an honest coin for two players, in which the probability of one of them winning is $\frac{1}{{\pi}^e}$.
2010 District Olympiad, 3
Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.
2009 AMC 12/AHSME, 15
Assume $ 0 < r < 3$. Below are five equations for $ x$. Which equation has the largest solution $ x$?
$ \textbf{(A)}\ 3(1 \plus{} r)^x \equal{} 7\qquad \textbf{(B)}\ 3(1 \plus{} r/10)^x \equal{} 7\qquad \textbf{(C)}\ 3(1 \plus{} 2r)^x \equal{} 7$
$ \textbf{(D)}\ 3(1 \plus{} \sqrt {r})^x \equal{} 7\qquad \textbf{(E)}\ 3(1 \plus{} 1/r)^x \equal{} 7$
KoMaL A Problems 2020/2021, A. 798
Let $0<p<1$ be given. Initially, we have $n$ coins, all of which have probability $p$ of landing on heads, and probability $1-p$ of landing on tails (the results of the tosses are independent of each other). In each round, we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that are needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]
1976 IMO Longlists, 20
Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and
\[a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots\]
Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$
\[|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,\]
and give one such $q$ explicitly.
2003 Iran MO (3rd Round), 24
$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.
2017 Iran MO (3rd round), 3
$30$ volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called [b]unusual[/b] if at the end of the league, the winner of the match have a smaller amount of wins than the loser of the match. A team is called [b]astonishing[/b] if all its matches are [b]unusual[/b] matches.
Find the maximum number of [b]astonishing[/b] teams.
2009 Tuymaada Olympiad, 2
An arrangement of chips in the squares of $ n\times n$ table is called [i]sparse[/i] if every $ 2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what $ n$ is this possible?
[i]Proposed by S. Berlov[/i]
1969 IMO Longlists, 24
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
LMT Speed Rounds, 2010.14
On the team round, an LMT team of six students wishes to divide itself into two distinct groups of three, one group to work on part $1,$ and one group to work on part $2.$ In addition, a captain of each group is designated. In how many ways can this be done?
2013 Kazakhstan National Olympiad, 2
Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$
2017 Ukrainian Geometry Olympiad, 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point of the circumcircle of $\Delta ABD$, different from the vertices. Line $PA$ intersects the line $CD$ at point $Q$. Let $O$ be the center of the circumcircle $\Delta PCQ$. Prove that $\angle ADO = 90^o$.