Found problems: 85335
2010 IMO Shortlist, 3
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
2023 Cono Sur Olympiad, 1
A list of \(n\) positive integers \(a_1, a_2,a_3,\ldots,a_n\) is said to be [i]good[/i] if it checks simultaneously:
\(\bullet a_1<a_2<a_3<\cdots<a_n,\)
\(\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.\)
For each \(n\ge 1\), determine how many [i]good[/i] lists of \(n\) numbers exist.
VMEO IV 2015, 11.1
Let $k \ge 0$ and $a, b, c$ be three positive real numbers such that $$\frac{a}{b}+\frac{b}{c}+ \frac{c}{a}= (k + 1)^2 + \frac{2}{k+ 1}.$$ Prove that $$a^2 + b^2 + c^2 \le (k^2 + 1)(ab + bc + ca).$$
2017 BMT Spring, 6
Given a cube with side length $ 1$, we perform six cuts as follows: one cut parallel to the $xy$-plane, two cuts parallel to the $yz$-plane, and three cuts parallel to the $xz$-plane, where the cuts are made uniformly independent of each other. What is the expected value of the volume of the largest piece?
2017 Korea National Olympiad, problem 4
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as
\[ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} \]
Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.
2022 JHMT HS, 6
For positive real numbers $a$ and $b,$ let $f(a,b)$ denote the real number $x$ such that area of the (non-degenerate) triangle with side lengths $a,b,$ and $x$ is maximized. Find
\[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]
1996 Flanders Math Olympiad, 4
Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$.
(a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$.
(b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$.
(c) Show, in (b) that $s=0$ implies $a_2=0$.
2008 Moldova MO 11-12, 2
Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.
1989 IMO Longlists, 41
Let $ f(x) \equal{} a \sin^2x \plus{} b \sin x \plus{} c,$ where $ a, b,$ and $ c$ are real numbers. Find all values of $ a, b$ and $ c$ such that the following three conditions are satisfied simultaneously:
[b](i)[/b] $ f(x) \equal{} 381$ if $ \sin x \equal{} \frac{1}{2}.$
[b](ii)[/b] The absolute maximum of $ f(x)$ is $ 444.$
[b](iii)[/b] The absolute minimum of $ f(x)$ is $ 364.$
1958 AMC 12/AHSME, 32
With $ \$1000$ a rancher is to buy steers at $ \$25$ each and cows at $ \$26$ each. If the number of steers $ s$ and the number of cows $ c$ are both positive integers, then:
$ \textbf{(A)}\ \text{this problem has no solution}\qquad\\
\textbf{(B)}\ \text{there are two solutions with }{s}\text{ exceeding }{c}\qquad \\
\textbf{(C)}\ \text{there are two solutions with }{c}\text{ exceeding }{s}\qquad \\
\textbf{(D)}\ \text{there is one solution with }{s}\text{ exceeding }{c}\qquad \\
\textbf{(E)}\ \text{there is one solution with }{c}\text{ exceeding }{s}$
2021 Math Prize for Girls Problems, 8
In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$,
what is the perimeter of $\triangle ABC$?
2022 IMO Shortlist, N8
Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.
2003 All-Russian Olympiad Regional Round, 10.6
Let $A_0$ be the midpoint of side $BC$ of triangle $ABC$, and $A'$ be the point of tangency with this side of the inscribed circle. Let's construct a circle $ \omega$ with center at $A_0$ and passing through $A'$. On other sides we will construct similar circles. Prove that if $ \omega$ is tangent to the cirucmscribed circle on arc $BC$ not containing $A$, then another one of the constructed circles touches the circumcircle.
1999 APMO, 1
Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.
2019 Junior Balkan MO, 4
A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.
2023 Bulgarian Autumn Math Competition, 8.4
In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for
$a)$ $k=5;$
$b)$ $k=8.$
1957 Miklós Schweitzer, 5
[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:
(a) $x,y,z$ are arbitrary non-zero real numbers;
(b) $a<x,y,z<b (1<a^{3}<b)$.
[b](R. 13)[/b]
2019 Malaysia National Olympiad, 4
Let $A=\{1,2,...,100\}$ and $f(k), k\in N$ be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k)=50$?
2004 Iran MO (2nd round), 4
$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that:
\[f(m)+f(n) \ | \ m+n .\]
2009 F = Ma, 2
Suppose that all collisions are instantaneous and perfectly elastic. After a long time, which of the following is true?
(A) The center block is moving to the left.
(B) The center block is moving to the right.
(C) The center block is at rest somewhere to the left of its initial position.
(D) The center block is at rest at its initial position.
(E) The center block is at rest somewhere to the right of its initial position.
1999 All-Russian Olympiad Regional Round, 9.2
In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.
2013 Czech-Polish-Slovak Junior Match, 4
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.
2023 Indonesia TST, 1
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2006 Purple Comet Problems, 13
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);
[/asy]
2014 Bundeswettbewerb Mathematik, 2
For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$.
Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation
\[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]