Found problems: 85335
2012 Brazil Team Selection Test, 4
Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
[i]Proposed by Canada[/i]
2021 Romania National Olympiad, 1
Let $ABC$ be an acute-angled triangle with the circumcenter $O$. Let $D$ be the foot of the altitude from $A$. If $OD\parallel AB$, show that $\sin 2B = \cot C$.
[i]Mădălin Mitrofan[/i]
2017 Federal Competition For Advanced Students, P2, 3
Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$.
Show that the sequence does not contain a square of a rational number.
Proposed by Theresia Eisenkölbl
1969 Bulgaria National Olympiad, Problem 2
Prove that
$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
2005 Paraguay Mathematical Olympiad, 3
The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$ Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .
2014 IFYM, Sozopol, 6
$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: \[ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}, \] \[ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} . \]
1999 AMC 12/AHSME, 15
Let $ x$ be a real number such that $ \sec x \minus{} \tan x \equal{} 2$. Then $ \sec x \plus{} \tan x \equal{}$
$ \textbf{(A)}\ 0.1 \qquad
\textbf{(B)}\ 0.2 \qquad
\textbf{(C)}\ 0.3 \qquad
\textbf{(D)}\ 0.4 \qquad
\textbf{(E)}\ 0.5$
2008 India Regional Mathematical Olympiad, 1
On a semicircle with diameter $AB$ and centre $S$, points $C$ and $D$ are given such that point $C$ belongs to arc $AD$. Suppose $\angle CSD = 120^\circ$. Let $E$ be the point of intersection of the straight lines $AC$ and $BD$ and $F$ the point of intersection of the straight lines $AD$ and $BC$. Prove that $EF=\sqrt{3}AB$.
2002 Moldova National Olympiad, 2
Does there exist a positive integer $ n>1$ such that $ n$ is a power of $ 2$ and one of the numbers obtained by permuting its (decimal) digits is a power of $ 3$ ?
2009 Miklós Schweitzer, 9
Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$
\[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]
1966 Swedish Mathematical Competition, 3
Show that an integer $= 7 \mod 8$ cannot be sum of three squares.
2014 ASDAN Math Tournament, 20
$ABCD$ is a parallelogram, and circle $S$ (with radius $2$) is inscribed insider $ABCD$ such that $S$ is tangent to all four line segments $AB$, $BC$, $CD$, and $DA$. One of the internal angles of the parallelogram is $60^\circ$. What is the maximum possible area of $ABCD$?
2021 AIME Problems, 4
There are real numbers $a, b, c, $ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i, $ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$
2007 Czech and Slovak Olympiad III A, 1
A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations:
[b](a)[/b] move the stone to a square that shares a common side with the square in which it stands;
[b](b)[/b] move it to a square sharing only one common vertex with the square in which it stands.
In addition, we are required that the first step must be [b](b)[/b]. Find all integers $n$ such that the stone can go through a certain path visiting every square exactly once.
MBMT Guts Rounds, 2015.4
Find the fourth-smallest positive integer that can be expressed as the product of two different prime numbers.
1995 AMC 8, 14
A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season?
$\text{(A)}\ 20 \qquad \text{(B)}\ 23 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$
2002 Austrian-Polish Competition, 5
Let $A$ be the set $\{2,7,11,13\}$. A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$. Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$.
1994 Abels Math Contest (Norwegian MO), 4a
In a group of $20$ people, each person sends a letter to $10$ of the others.
Prove that there are two persons who send a letter to each other.
2022 Kyiv City MO Round 1, Problem 1
What's the smallest possible value of $$\frac{(x+y+|x-y|)^2}{xy}$$ over positive real numbers $x, y$?
2007 Purple Comet Problems, 12
Find the maximum possible value of $8\cdot 27^{\log_6 x}+27\cdot 8^{\log_6 x}-x^3$ as $x$ varies over the positive real numbers.
2010 Romania National Olympiad, 3
Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.
2010 Stanford Mathematics Tournament, 2
Find the smallest prime $p$ such that the digits of $p$ (in base 10) add up to a prime number greater than $10$.
2011 Korea Junior Math Olympiad, 7
For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of
\[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]
2008 Tuymaada Olympiad, 8
A convex hexagon is given. Let $ s$ be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed $ s.$
[i]Author: N. Sedrakyan[/i]
2019 Serbia JBMO TST, 4
$4.$ On a table there are notes of values: $1$, $2$, $5$, $10$, $20$ ,$50$, $100$, $200$, $500$, $1000$, $2000$ and $5000$ (the number of any of these notes can be any non-negative integer). Two players , First and Second play a game in turns (First plays first). With one move a player can take any one note of value higher than $1$ , and replace it with notes of less value. The value of the chosen note is equal to the sum of the values of the replaced notes. The loser is the player which can not play any more moves. Which player has the winning strategy?