Found problems: 85335
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?
1963 Swedish Mathematical Competition., 1
How many positive integers have square less than $10^7$?
1983 Tournament Of Towns, (047) 4
$a_1,a_2,a_3,...$ is a monotone increasing sequence of natural numbers. It is known that for any $k, a_{a_k} = 3k$.
a) Find $a_{100}$.
b) Find $a_{1983}$.
(A Andjans, Riga)
PS. (a) for Juniors, (b) for Seniors
2011 China Team Selection Test, 1
In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.
2016 Azerbaijan BMO TST, 4
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2014 Moldova Team Selection Test, 2
Let $a,b,c$ be positive real numbers such that $abc=1$. Determine the minimum value of
$E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}$ .
1998 Putnam, 2
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists.
2012 AMC 12/AHSME, 15
A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
$ \textbf{(A)}\ \dfrac{49}{512}
\qquad\textbf{(B)}\ \dfrac{7}{64}
\qquad\textbf{(C)}\ \dfrac{121}{1024}
\qquad\textbf{(D)}\ \dfrac{81}{512}
\qquad\textbf{(E)}\ \dfrac{9}{32}
$
2022 Azerbaijan BMO TST, N4*
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that
$$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$
holds for all $m \in \mathbb{Z}$.
2019 India PRMO, 24
For $n \geq 1$, let $a_n$ be the number beginning with $n$ $9$'s followed by $744$; eg., $a_4=9999744$. Define $$f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \}$$, for $n\geq 1$. Find $f(1)+f(2)+f(3)+ \cdots + f(10)$.
1957 AMC 12/AHSME, 25
The vertices of triangle $ PQR$ have coordinates as follows: $ P(0,a),\,Q(b,0),\,R(c,d),$ where $ a,\,b,\,c$ and $ d$ are positive. The origin and point $ R$ lie on opposite sides of $ PQ$. The area of triangle $ PQR$ may be found from the expression:
$ \textbf{(A)}\ \frac{ab \plus{} ac \plus{} bc \plus{} cd}{2} \qquad
\textbf{(B)}\ \frac{ac \plus{} bd \minus{} ab}{2}\qquad
\textbf{(C)}\ \frac{ab \minus{} ac \minus{} bd}{2}\qquad
\textbf{(D)}\ \frac{ac \plus{} bd \plus{} ab}{2}\qquad
\textbf{(E)}\ \frac{ac \plus{} bd \minus{} ab \minus{} cd}{2}$
2015 Kyiv Math Festival, P3
Is it true that every positive integer greater than 30 is a sum of 4 positive integers such that each two of them have a common divisor greater than 1?
2015 Mathematical Talent Reward Programme, SAQ: P 5
Let $a$ be the smallest and $A$ the largest of $n$ distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to $n a$ and that the greatest common divisor is less than or equal to $\frac{A}{n}$
Russian TST 2015, P3
Given two integers $h \geq 1$ and $p \geq 2$, determine the minimum number of pairs of opponents an $hp$-member parliament may have, if in every partition of the parliament into $h$ houses of $p$ member each, some house contains at least one pair of opponents.