Found problems: 85335
2019 Sharygin Geometry Olympiad, 6
Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic
2009 Stanford Mathematics Tournament, 4
How many ways are there to write $657$ as a sum of powers of two where each power of two is used at
most twice in the sum? For example, $256+256+128+16+1$ is a valid sum.
2021 Estonia Team Selection Test, 1
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom
2018 IFYM, Sozopol, 8
Find all positive integers $n$ for which a square[b][i] n x n[/i][/b] can be covered with rectangles [b][i]k x 1[/i][/b] and one square [b][i]1 x 1[/i][/b] when:
a) $k = 4$ b) $k = 8$
2020 Peru IMO TST, 6
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
2006 Cezar Ivănescu, 3
[b]a)[/b] Let be a sequence $ \left( x_n \right)_{n\ge 1} $ defined by the recursion $ x_{n+1}=\frac{1+x_n}{1-x_n} , $ with $ x_1=2006. $ Calculate $ \lim_{n\to\infty } \frac{x_1+x_2+\cdots +x_n}{n} . $
[b]b)[/b] Prove that if a convergent sequence $ \left( s_n \right)_{n\ge 1} $ verifies $ a_{2^n} =na_n , $ for any natural numbers $ n, $ then $ a_n=0, $ for any natural numbers $ n. $
[i]Cornel Stoicescu[/i]
1994 IMO, 1
Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that
\[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}.
\]
1978 IMO Longlists, 39
$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.
2024 IFYM, Sozopol, 7
Consider a finite undirected graph in which each edge belongs to at most three cycles. Prove that its vertices can be colored with three colors so that any two vertices connected by an edge have different colors.
[i](A cycle in a graph is a sequence of distinct vertices \( v_1, v_2, \ldots, v_k \), \( k \geq 3 \), such that \( v_i v_{i+1} \) is an edge for each \( i = 1, 2, \ldots, k \); we consider \( v_{k+1} = v_1 \). The edges \( v_i v_{i+1} \) belong to the cycle.)[/i]
2023 Romania National Olympiad, 3
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
2018 German National Olympiad, 3
Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes.
Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions:
1) There are no two red boxes in the same column or in the same row.
2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column.
[i]Proposed by Christian Reiher[/i]
2003 IMC, 1
Let $A,B \in \mathbb{R}^{n\times n}$ such that $AB+B+A=0$. Prove that $AB=BA$.
2010 Math Prize For Girls Problems, 1
If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$, compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
- \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
\right).
\]
2007 Purple Comet Problems, 12
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.
1962 AMC 12/AHSME, 40
The limiting sum of the infinite series, $ \frac{1}{10} \plus{} \frac{2}{10^2} \plus{} \frac{3}{10^3} \plus{} \dots$ whose $ n$th term is $ \frac{n}{10^n}$ is:
$ \textbf{(A)}\ \frac19 \qquad
\textbf{(B)}\ \frac{10}{81} \qquad
\textbf{(C)}\ \frac18 \qquad
\textbf{(D)}\ \frac{17}{72} \qquad
\textbf{(E)}\ \text{larger than any finite quantity}$
2016 ASDAN Math Tournament, 13
Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)
2019 Moroccan TST, 2
Let $a>1$ be a real number. Prove that for all $n\in\mathbb{N}*$ that :
$\frac{a^n-1}{n}\ge \sqrt{a}^{n+1}-\sqrt{a}^{n-1}$
2024 German National Olympiad, 5
Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
1984 National High School Mathematics League, 4
The number of real roots of the equation $\sin x=\lg x$ is
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}$more than $3$
1954 Putnam, B1
Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.
2002 AMC 12/AHSME, 12
For how many integers $ n$ is $ \frac{n}{20\minus{}n}$ the square of an integer?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 10$
2002 IMC, 12
Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function whose gradient $\nabla f$
exists at every point of $\mathbb{R}^{n}$ and satisfies the condition
$$\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.$$
Prove that
$$ \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle. $$
2021 IMO Shortlist, A2
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2021 Kyiv City MO Round 1, 8.3
The $1 \times 1$ cells located around the perimeter of a $3 \times 3$ square are filled with the numbers $1,
2, \ldots, 8$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $8$, and in the upper left is the number $6$ (see the figure below).
[img]https://i.ibb.co/bRmd12j/Kyiv-MO-2021-Round-1-8-2.png[/img]
How many different ways to fill the remaining cells are there under these conditions?
[i]Proposed by Mariia Rozhkova[/i]
2015 European Mathematical Cup, 2
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$
[i]Dimitar Trenevski[/i]