Found problems: 85335
2018 Argentina National Olympiad, 3
You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box.
a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img]
b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.
2011 Today's Calculation Of Integral, 754
Let $S_n$ be the area of the figure enclosed by a curve $y=x^2(1-x)^n\ (0\leq x\leq 1)$　and the $x$-axis.
Find $\lim_{n\to\infty} \sum_{k=1}^n S_k.$
2013 Korea Junior Math Olympiad, 8
Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.
2011 Sharygin Geometry Olympiad, 5
The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.
2014 Sharygin Geometry Olympiad, 11
Points $K, L, M$ and $N$ lying on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$.
2015 239 Open Mathematical Olympiad, 4
ÙŽA natural number $n$ is given. Let $f(x,y)$ be a polynomial of degree less than $n$ such that for any positive integers $x,y\leq n, x+y \leq n+1$ the equality $f(x,y)=\frac{x}{y}$ holds. Find $f(0,0)$.
2008 Iran MO (3rd Round), 1
Prove that the number of pairs $ \left(\alpha,S\right)$ of a permutation $ \alpha$ of $ \{1,2,\dots,n\}$ and a subset $ S$ of $ \{1,2,\dots,n\}$ such that
\[ \forall x\in S: \alpha(x)\not\in S\]
is equal to $ n!F_{n \plus{} 1}$ in which $ F_n$ is the Fibonacci sequence such that $ F_1 \equal{} F_2 \equal{} 1$
2016 China Northern MO, 8
Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$.
$A\subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$, there exists $i,j(1\leq i<j\leq4)$, $(x_i-x_j)(y_i-y_j)<0$. Find the maximum value of $|A|$.
2013 NIMO Problems, 9
Haddaway once asked,``what is love?''. The answer can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are positive integers such that $m^2 + n^2 < 2013$. Find $100m+n$.
[i]Proposed by Evan Chen[/i]
2013 All-Russian Olympiad, 3
Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one).
2007 Indonesia MO, 5
Let $ r$, $ s$ be two positive integers and $ P$ a 'chessboard' with $ r$ rows and $ s$ columns. Let $ M$ denote the maximum value of rooks placed on $ P$ such that no two of them attack each other.
(a) Determine $ M$.
(b) How many ways to place $ M$ rooks on $ P$ such that no two of them attack each other?
[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
2022 BMT, 9
We define a sequence $x_1 = \sqrt{3}, x_2 =-1, x_3 =2 - \sqrt{3},$ and for all $n \geq 4$
$$(x_n + x_{n-3})(1 - x^2_{n-1}x^2_{n-2}) = 2x_{n-1}(1 + x^2_{n-2}).$$
Suppose $m$ is the smallest positive integer for which $x_m$ is undefined. Compute $m.$
2013 AMC 12/AHSME, 19
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $
2016 Belarus Team Selection Test, 3
Point $A,B$ are marked on the right branch of the hyperbola $y=\frac{1}{x},x>0$. The straight line $l$ passing through the origin $O$ is perpendicular to $AB$ and meets $AB$ and given branch of the hyperbola at points $D$ and $C$ respectively. The circle through $A,B,C$ meets $l$ at $F$.
Find $OD:CF$
2009 Romania National Olympiad, 4
Find all functions $ f:[0,1]\longrightarrow [0,1] $ that are bijective, continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow\mathbb{R} , $ the following equality holds.
$$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$
Ukrainian TYM Qualifying - geometry, IV.10
Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle.
i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$
($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$).
ii). Find out when equality is achieved.
2009 Balkan MO, 3
A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled
i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$,
ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry?
[i]Bulgaria[/i]
2016 Online Math Open Problems, 26
Let $S$ be the set of all pairs $(a, b)$ of integers satisfying $0 \le a, b \le 2014.$ For any pairs $s_1 = (a_1, b_1), s_2 = (a_2, b_2) \in S$, define \[s_1 + s_2 = ((a_1 + a_2)_{2015}, (b_1 + b_2)_{2015}) \\ \text { and } \\ s_1 \times s_2 = ((a_1a_2 + 2b_1b_2)_{2015}, (a_1b_2 + a_2b_1)_{2015}), \] where $n_{2015}$ denotes the remainder when an integer $n$ is divided by $2015.$
Compute the number of functions $f : S \rightarrow S$ satisfying \[ f(s_1 + s_2) = f(s_1) + f(s_2) \text{ and } f(s_1 \times s_2) = f(s_1) \times f(s_2) \] for all $s_1, s_2 \in S.$
[i] Proposed by Yang Liu [/i]
2020 Czech and Slovak Olympiad III A, 6
For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs.
(Note: A positive integer cannot begin with a digit of $0$.)
(Jaromir Simsa)
2021 Iran RMM TST, 2
In a chess board we call a group of queens [i]independant[/i] if no two are threatening each other. In an $n$ by $n$ grid, we put exaxctly one queen in each cell ofa greed. Let us denote by $M_n$ the minimum number of independant groups that hteir union contains all the queens. Let $k$ be a positive integer, prove that $M_{3k+1} \le 3k+2$
Proposed by [i]Alireza Haghi[/i]
2010 China Team Selection Test, 2
Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose
\[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\]
holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.
2023 Dutch BxMO TST, 2
Find all functions $f : \mathbb R \to \mathbb R$ for which
\[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\]
for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!
Russian TST 2015, P4
Let $G$ be a tournoment such that it's edges are colored either red or blue.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
2009 All-Russian Olympiad, 5
Let $ a$, $ b$, $ c$ be three real numbers satisfying that \[ \left\{\begin{array}{c c c} \left(a\plus{}b\right)\left(b\plus{}c\right)\left(c\plus{}a\right)&\equal{}&abc\\ \left(a^3\plus{}b^3\right)\left(b^3\plus{}c^3\right)\left(c^3\plus{}a^3\right)&\equal{}&a^3b^3c^3\end{array}\right.\] Prove that $ abc\equal{}0$.
1986 IMO Longlists, 55
Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ , $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$, $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$