Found problems: 85335
2024 Bosnia and Herzegovina Junior BMO TST, 4.
Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this:
[b]- [/b]a white square becomes black;
[b]-[/b]a black square becomes blue;
[b]-[/b]a blue square becomes white.
For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.
2015 BAMO, 2
Members of a parliament participate in various committees. Each committee consists of at least $2$ people, and it is known that every two committees have at least one member in common. Prove that it is possible to give each member a colored hat (hats are available in black, white or red) so that every committee contains at least $2$ members with different hat colors.
2012 Vietnam National Olympiad, 1
Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit.
2006 AMC 10, 5
A 2 x 3 rectangle and a 3 x 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
$ \textbf{(A) } 16 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 49 \qquad \textbf{(E) } 64$
2010 Indonesia TST, 1
find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \]
for every natural numbers $ a $ and $ b $
1970 Vietnam National Olympiad, 2
Let $N=1890*1930*1970$, find the number of divisors of N which are not divisors of $45$
2013 Kosovo National Mathematical Olympiad, 4
Calculate:
$\sqrt{3\sqrt{5\sqrt{3\sqrt{5...}}}}$
CIME I 2018, 12
Define a permutation of the set $\{1,2,3,...,n\}$ to be $\textit{sortable}$ if upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when $$\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)$$ is divided by $1000$. Note that the empty set is considered sortable.
[i]Proposed by [b]FedeX333X[/b][/i]
2014 Dutch IMO TST, 2
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
2016 Miklós Schweitzer, 2
Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.
2018 Benelux, 1
(a) Determine the minimal value of
$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $
where $x$ and $y$ vary over the positive reals.
(b) Determine the minimal value of
$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $
where $x$ and $y$ vary over the positive reals.
2011 International Zhautykov Olympiad, 1
Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively.
a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$.
b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?
2005 India IMO Training Camp, 3
A merida path of order $2n$ is a lattice path in the first quadrant of $xy$- plane joining $(0,0)$ to $(2n,0)$ using three kinds of steps $U=(1,1)$, $D= (1,-1)$ and $L= (2,0)$, i.e. $U$ joins $x,y)$ to $(x+1,y+1)$ etc... An ascent in a merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of merdia paths of order $2n$ with exactly $k$ ascents, compute $S(n,1)$ and $S(n,n-1)$.
2021 CMIMC, 2.4
What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers?
[i]Proposed by Dilhan Salgado[/i]
2020 Australian Maths Olympiad, 7
A $\emph{tetromino tile}$ is a tile that can be formed by gluing together four unit square tiles, edge to edge. For each positive integer $\emph{n}$, consider a bathroom whose floor is in the shape of a $2\times2 n$ rectangle. Let $T_n$ be the number of ways to tile this bathroom floor with tetromino tiles. For example, $T_2 = 4$ since there are four ways to tile a $2\times4$ rectangular bathroom floor with tetromino tiles, as shown below.
[click for diagram]
Prove that each of the numbers $T_1, T_2, T_3, ...$ is a perfect square.
2003 Bosnia and Herzegovina Team Selection Test, 3
Prove that for every positive integer $n$ holds:
$(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$
2001 China Western Mathematical Olympiad, 1
The sequence $ \{x_n\}$ satisfies $ x_1 \equal{} \frac {1}{2}, x_{n \plus{} 1} \equal{} x_n \plus{} \frac {x_n^2}{n^2}$. Prove that $ x_{2001} < 1001$.
II Soros Olympiad 1995 - 96 (Russia), 9.9
There are $5$ ingots weighing $1$, $2$, $3$, $4$ and $5$ kg with an unknown copper content that varies in different ingots. Each ingot must be divided into $5$ parts and $5$ new ingots of the same mass of $1$, $2$, $3$, $4$ and $5$ kg must be made. This requires that the percentage of copper in all pieces be the same, regardless of what it was in the original pieces. What parts should each piece be divided into?
2023 Sinapore MO Open, P2
A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false?
(1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$.
(2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.
2015 Switzerland - Final Round, 8
Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.
1996 South africa National Olympiad, 2
Find all real numbers for which $3^x+4^x=5^x$.
2022-23 IOQM India, 17
For a positive integer $n>1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$. For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$. Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$. Find the largest integer not exceeding $\sqrt{N}$
2022 Utah Mathematical Olympiad, 4
Alpha and Beta are playing a game on a $10\times 100$ grid of squares. At each turn, they can fold the grid along any of the interior horizontal or vertical gridlines, which creates a smaller (folded) grid of squares (on the first move, they can choose one of $9$ horizontal or $99$ vertical gridlines). The person who makes the last fold wins. If both players play optimally and Alpha starts, determine with proof who wins.
1999 IMO Shortlist, 6
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
2022 Serbia National Math Olympiad, P4
Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$, $n\in\mathbb{N}$, such that $gcd(x, n)$ is either $1$ or prime. Prove
$$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$
For which $n$ does equality hold?