Found problems: 85335
2014 Puerto Rico Team Selection Test, 6
Natural numbers are written in the cells of of a $2014\times2014$ regular square grid such that every number is the average of the numbers in the adjacent cells. Describe and prove how the number distribution in the grid can be.
KoMaL A Problems 2020/2021, A. 790
Andrew and Barry play the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k,$ and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game.
Which player has a winning strategy?
[i]Submitted by András Imolay, Budapest[/i]
2002 Croatia Team Selection Test, 2
A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$
2015 ASDAN Math Tournament, 10
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?
2017 India IMO Training Camp, 3
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either [i]on[/i] or [i]off[/i]. Every second, each lamp undergoes a change according to the following rule:
(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is [i]off[/i] right now. (Indices taken mod $n$.)
(b) Otherwise, $L_i$ is [i]on[/i] right now.
Initially, all the lamps are [i]off[/i], except for $L_1$ which is [i]on[/i]. Prove that for infinitely many integers $n$ all the lamps will be [i]off[/i] eventually, after a finite amount of time.
2014 Contests, 3
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
2018 Harvard-MIT Mathematics Tournament, 8
For how many pairs of sequences of nonnegative integers $(b_1,b_2,\ldots, b_{2018})$ and $(c_1,c_2,\ldots, c_{2018})$ does there exist a sequence of nonnegative integers $(a_0,\ldots, a_{2018})$ with the following properties:
[list]
[*] For $0\leq i\leq 2018,$ $a_i<2^{2018}.$
[*] For $1\leq i \leq 2018, b_i=a_{i-1}+a_i$ and $c_i=a_{i-1}|a_i$;
[/list]
where $|$ denotes the bitwise or operation?
2011 AMC 10, 8
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
$ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 50\qquad\textbf{(E)}\ 60 $
II Soros Olympiad 1995 - 96 (Russia), 10.9
The opposite sides of a quadrilateral inscribed in a circle intersect at points $K$ and $L$. Let $F$ be the midpoint of $KL$, $E$ and $G$ be the midpoints of the diagonals of the given quadrilateral. It is known that $FE = a$, $FG = b$. Calculate $KL$ in terms of $a$ and $b.$
(It is known that the points $F$, $E$ and $G$ lie on the same straight line. This is true for any quadrilateral, not necessarily inscribed. The indicated straight line is sometimes called the Newton−Gauss line. This fact can be used without proof in proving the problem, as it is known).
2021 MOAA, 9
Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
[i]Proposed by Andy Xu[/i]
2006 Kazakhstan National Olympiad, 5
Prove that for every $ x $ such that $ \sin x \neq 0 $, exists natural $ n $ such that $ | \sin nx | \geq \frac {\sqrt {3}} {2} $.
2007 Junior Macedonian Mathematical Olympiad, 3
Let $a$, $b$, $c$ be real numbers such that $0 < a \le b \le c$. Prove that
$(a + 3b)(b + 4c)(c + 2a) \ge 60abc$.
When does equality hold?
2013 Kazakhstan National Olympiad, 3
How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right?
2010 NZMOC Camp Selection Problems, 5
The diagonals of quadrilateral $ABCD$ intersect in point $E$. Given that $|AB| =|CE|$, $|BE| = |AD|$, and $\angle AED = \angle BAD$, determine the ratio $|BC|:|AD|$.
1976 IMO Shortlist, 12
The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.
Champions Tournament Seniors - geometry, 2011.4
The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).
2007 IMO Shortlist, 7
Let $ n$ be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$.
[i]Author: Gerhard Wöginger, Netherlands [/i]
2013 BMT Spring, 5
Consider the roots of the polynomial $x^{2013}-2^{2013}=0$. Some of these roots also satisfy $x^k-2^k=0$, for some integer $k<2013$. What is the product of this subset of roots?
Russian TST 2021, P3
Given a natural number $n\geqslant 2$, find the smallest possible number of edges in a graph that has the following property: for any coloring of the vertices of the graph in $n{}$ colors, there is a vertex that has at least two neighbors of the same color as itself.
2024 HMNT, 4
Compute the number of ways to pick a three-element subset of $$\{10^1+1, 10^2+1, 10^3+1, 10^4+1, 10^5+1, 10^6+1, 10^7+1\}$$ such that the product of the $3$ numbers in the subset has no digits besides $0$ and $1$ when written in base $10.$
2017 Yasinsky Geometry Olympiad, 3
The two sides of the triangle are $10$ and $15$. Prove that the length of the bisector of the angle between them is less than $12$.
2018 USAJMO, 2
Let \(a,b,c\) be positive real numbers such that \(a+b+c=4\sqrt[3]{abc}\). Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]
2016 Peru Cono Sur TST, P3
Ten students are seated around a circular table. The teacher has a list of fifteen problems and each student is given six problems, in such a way that each problem is given exactly four times and any two students they have at most three problems in common. Prove that no matter how the teacher distributes the problems, there will always be two students sitting next to each other who have at least one problem in common.
2001 India National Olympiad, 6
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$
1999 Romania National Olympiad, 3
Let $a,b \in \mathbb{R},$ $a<b$ and $f,g:[a,b] \to \mathbb{R}$ two differentiable functions with increasing derivatives and $f'(a)>0,$ $g'(a)>0.$ Prove that there exists $c \in [a,b]$ such that $$\frac{f(b)-f(a)}{b-a} \cdot \frac{g(b)-g(a)}{b-a}=f'(c)g'(c).$$