Found problems: 85335
1994 Bundeswettbewerb Mathematik, 2
Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”
2024 ITAMO, 5
A [i]fortress[/i] is a finite collection of cells in an infinite square grid with the property that one can pass from any cell of the fortress to any other by a sequence of moves to a cell with a common boundary line (but it can have "holes").
The [i]walls[/i] of a fortress are the unit segments between cells belonging to the fortress and cells not belonging to the fortress.
The [i]area[/i] $A$ of a fortress is the number of cells it consists of. The [i]perimeter[/i] $P$ is the total length of its walls.
Each cell of the fortress can contain a [i]guard[/i] which can oversee the cells to the top, the bottom, the right and the left of this cell, up until the next wall (it also oversees its own cell).
(a) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of perimeter $P \le 2024$.
(b) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of area $A \le 2024$.
1997 Korea - Final Round, 3
Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that
[list]
(i) if $ x < y$, then $ f(x) < f(y)$;
(ii) $ f(xy) \equal{} g(y)f(x) \plus{} f(y)$ for all $ x, y \in \mathbb R$.
[/list]
1994 AIME Problems, 1
The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?
2003 Italy TST, 3
Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.
LMT Speed Rounds, 19
Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$?
[i]Proposed by Evin Liang[/i]
2010 South africa National Olympiad, 4
Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that
\[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]
2013 Germany Team Selection Test, 2
Given a $m\times n$ grid rectangle with $m,n \ge 4$ and a closed path $P$ that is not self intersecting from inner points of the grid, let $A$ be the number of points on $P$ such that $P$ does not turn in them and let $B$ be the number of squares that $P$ goes through two non-adjacent sides of them furthermore let $C$ be the number of squares with no side in $P$. Prove that $$A=B-C+m+n-1.$$
2014 ISI Entrance Examination, 4
Let $f,g$ are defined in $(a,b)$ such that $f(x),g(x)\in\mathcal{C}^2$ and non-decreasing in an interval $(a,b)$ . Also suppose $f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)$. Also it is given that $f(x)g(x)$ is linear in $(a,b)$. Show that $f\equiv 0 \text{ and } g\equiv 0$ in $(a,b)$.
2000 Czech and Slovak Match, 6
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.
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.
2018 District Olympiad, 3
Show that a continuous function $f : \mathbb{R} \to \mathbb{R}$ is increasing if and only if
\[(c - b)\int_a^b f(x)\, \text{d}x \le (b - a) \int_b^c f(x) \, \text{d}x,\]
for any real numbers $a < b < c$.
2004 AIME Problems, 11
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2024 APMO, 4
Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.
2006 Baltic Way, 2
Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$
Prove that
\[a_1^2+a_2^2+\ldots+a_{59}^2\le 2006\]
2011 Iran MO (3rd Round), 7
Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting.
[i]proposed by Sepehr Ghazi-Nezami[/i]
2003 Silk Road, 1
Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number.
Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive.
Prove, that if exist at least one $leading$ element, then sum of all $leading$'s elements is positive.
Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]
2022 Stars of Mathematics, 2
Given are real numbers $a_1, a_2, \ldots, a_n$ ($n>3$), such that $a_k^3=a_{k+1}^2+a_{k+2}^2+a_{k+3}^2$ for all $k=1,2,...,n$. Prove that all numbers are equal.
2005 Germany Team Selection Test, 2
Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases
[b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$.
[b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$.
In other words, find the Steiner trees of the set $M$ in the above two cases.
2009 Romania National Olympiad, 3
Let be a natural number $ n, $ a permutation $ \sigma $ of order $ n, $ and $ n $ nonnegative real numbers $ a_1,a_2,\ldots , a_n. $ Prove the following inequality.
$$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$
2018 PUMaC Live Round, 4.2
Some number of regular polygons meet at a point on the plane such that the polygons' interiors do not overlap, but the polygons fully surround the point (i.e. a sufficiently small circle centered at the point would be contained in the union of the polygons). What is the largest possible number of sides in any of the polygons?
2007 Oral Moscow Geometry Olympiad, 1
Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut).
(V. Shevyakov)
2013 AMC 8, 14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$
1978 Romania Team Selection Test, 2
Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.
2012 Philippine MO, 5
There are exactly $120$ Twitter subscribers from National Science High School. Statistics show that each of $10$ given celebrities has at least $85$ followers from National Science High School. Prove that there must be two students such that each of the $10$ celebrities is being followed in Twitter by at least one of these students.