Found problems: 85335
1995 AMC 12/AHSME, 10
The area of the triangle bounded by the lines $y = x, y = -x$ and $y = 6$ is
$
\mathbf{(A)}\; 12\qquad
\mathbf{(B)}\; 12\sqrt2\qquad
\mathbf{(C)}\; 24\qquad
\mathbf{(D)}\; 24\sqrt2\qquad
\mathbf{(E)}\; 36$
1966 IMO Shortlist, 38
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
2024 Switzerland Team Selection Test, 7
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.
[i]Thanasin Nampaisarn, Thailand[/i]
2008 Cono Sur Olympiad, 4
What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?
2010 Slovenia National Olympiad, 1
Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.
1986 Tournament Of Towns, (121) 3
A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins?
{S. Fomin , Leningrad)
1975 AMC 12/AHSME, 25
A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player?
$ \textbf{(A)}\ \text{the woman} \qquad\textbf{(B)}\ \text{her son} \qquad\textbf{(C)}\ \text{her brother} \qquad\textbf{(D)}\ \text{her daughter} \\ \qquad\textbf{(E)}\ \text{No solution is consistent with the given information} $
1982 Austrian-Polish Competition, 3
If $n \ge 2$ is an integer, prove the equality
$$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$
2019 Sharygin Geometry Olympiad, 3
Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.
2025 Belarusian National Olympiad, 10.7
For every positive integer $n$ write all its divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$.
Find all $n$ such that $2025 \cdot n=d_{20} \cdot d_{25}$.
[i]I. Voronovich[/i]
1985 AMC 12/AHSME, 20
A wooden cube with edge length $ n$ units (where $ n$ is an integer $ >2$) is painted black all over. By slices parallel to its faces, the cube is cut into $ n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $ n$?
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ \text{none of these}$
2025 Azerbaijan IZhO TST, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that
$$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$
for any $x;y\in\mathbb{Q}$
1999 Baltic Way, 16
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.
2004 Switzerland Team Selection Test, 2
Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.
2014 Bosnia And Herzegovina - Regional Olympiad, 1
Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$ where $x$, $y$ and $z$ are integers
2014 Contests, 4
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.
1998 Tournament Of Towns, 3
Nine numbers are arranged in a square table:
$a_1 \,\,\, a_2 \,\,\,a_3$
$b_1 \,\,\,b_2 \,\,\,b_3$
$c_1\,\,\, c_2 \,\,\,c_3$ .
It is known that the six numbers obtained by summing the rows and columns of the table are equal:
$a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ .
Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns:
$a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ .
(V Proizvolov)
2015 Benelux, 2
Let $ABC$ be an acute triangle with circumcentre $O$. Let $\mathit{\Gamma}_B$ be the circle through $A$ and $B$ that is tangent to $AC$, and let $\mathit{\Gamma}_C$ be the circle through $A$ and $C$ that is tangent to $AB$. An arbitrary line through $A$ intersects $\mathit{\Gamma}_B$ again in $X$ and $\mathit{\Gamma}_C$ again in $Y$. Prove that $|OX|=|OY|$.
2012 EGMO, 5
The numbers $p$ and $q$ are prime and satisfy
\[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\]
for some positive integer $n$. Find all possible values of $q-p$.
[i]Luxembourg (Pierre Haas)[/i]
1956 Polish MO Finals, 2
Prove that if $$ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a + b + c}$$ and $ n $ is any odd natural number, then
$$ \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n} =\frac{1}{a^n + b^n + c^n}$$
1968 Leningrad Math Olympiad, 8.6*
All $10$-digit numbers consisting of digits $1, 2$ and $3$ are written one under the other. Each number has one more digit added to the right. $1$, $2$ or $3$, and it turned out that to the number $111. . . 11$ added $1$ to the number $ 222. . . 22$ was assigned $2$, and the number $333. . . 33$ was assigned $3$. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.
2009 Kazakhstan National Olympiad, 5
Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$.
Proved that $ \angle AKP = \angle PKC$.
As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)
1994 All-Russian Olympiad Regional Round, 10.7
In a convex pentagon $ ABCDE$ side $ AB$ is perpendicular to $ CD$ and side $ BC$ is perpendicular to $ DE$. Prove that if $ AB \equal{} AE \equal{} ED \equal{} 1$, then
$ BC \plus{} CD < 1$.
2024 Ukraine National Mathematical Olympiad, Problem 6
You are given a convex hexagon with parallel opposite sides. For each pair of opposite sides, a line is drawn parallel to these sides and equidistant from them. Prove that the three lines thus obtained intersect at one point if and only if the lengths of the opposite sides are equal.
[i]Proposed by Nazar Serdyuk[/i]
1984 Putnam, B1
Let $n$ be a positive integer, and define $f(n)=1!+2!+\ldots+n!$. Find polynomials $P$ and $Q$ such that
$$f(n+2)=P(n)f(n+1)+Q(n)f(n)$$for all $n\ge1$.