Found problems: 85335
2003 Balkan MO, 4
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]
1991 All Soviet Union Mathematical Olympiad, 553
The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.
1992 IMO Longlists, 47
Evaluate
\[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]
2017 Bosnia And Herzegovina - Regional Olympiad, 3
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares
2020 AMC 10, 6
Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome—it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?
$\textbf{(A) }50 \qquad \textbf{(B) }55 \qquad \textbf{(C) }60\qquad \textbf{(D) }65\qquad\textbf{(E) }70$
2017 Math Hour Olympiad, 8-10
[u]Round 1[/u]
[b]p1. [/b]The Queen of Bees invented a new language for her hive. The alphabet has only $6$ letters: A, C, E, N, R, T; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the word TRANCE immediately follows NECTAR. What is the last word in the dictionary?
[b]p2.[/b] Is it possible to solve the equation $\frac{1}{x}= \frac{1}{y} +\frac{1}{z}$ with $x,y,z$ integers (positive or negative) such that one of the numbers $x,y,z$ has one digit, another has two digits, and the remaining one has three digits?
[b]p3.[/b] The $10,000$ dots in a $100\times 100$ square grid are all colored blue. Rekha can paint some of them red, but there must always be a blue dot on the line segment between any two red dots. What is the largest number of dots she can color red? The picture shows a possible coloring for a $5\times 7$ grid.
[img]https://cdn.artofproblemsolving.com/attachments/0/6/795f5ab879938ed2a4c8844092b873fb8589f8.jpg[/img]
[b]p4.[/b] Six flies rest on a table. You have a swatter with a checkerboard pattern, much larger than the table. Show that there is always a way to position and orient the swatter to kill at least five of the flies. Each fly is much smaller than a swatter square and is killed if any portion of a black square hits any part of the fly.
[b]p5.[/b] Maryam writes all the numbers $1-81$ in the cells of a $9\times 9$ table. Tian calculates the product of the numbers in each of the nine rows, and Olga calculates the product of the numbers in every column. Could Tian's and Olga's lists of nine products be identical?
[u]Round 2[/u]
[b]p6.[/b] A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. The figure shows a particular set of $4$ points and demonstrates that it is epic. What is the maximum possible size of an epic set?
[img]https://cdn.artofproblemsolving.com/attachments/e/f/44fd1679c520bdc55c78603190409222d0b721.jpg[/img]
[b]p7.[/b] Froggy Chess is a game played on a pond with lily pads. First Judit places a frog on a pad of her choice, then Magnus places a frog on a different pad of his choice. After that, they alternate turns, with Judit moving first. Each player, on his or her turn, selects either of the two frogs and another lily pad where that frog must jump. The jump must reduce the distance between the frogs (all distances between the lily pads are different), but both frogs cannot end up on the same lily pad. Whoever cannot make a move loses. The picture below shows the jumps permitted in a particular situation.
Who wins the game if there are $2017$ lily pads?
[img]https://cdn.artofproblemsolving.com/attachments/a/9/1a26e046a2a614a663f9d317363aac61654684.jpg[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 Contests, 2
Find all prime numbers $p, q, r$ such that
\[15p+7pq+qr=pqr.\]
1982 All Soviet Union Mathematical Olympiad, 340
The square table $n\times n$ is filled by integers. If the fields have common side, the difference of numbers in them doesn't exceed $1$. Prove that some number is encountered not less than
a) not less than $[n/2]$ times ($[ ...]$ mean the whole part),
b) not less than $n$ times.
2004 AIME Problems, 10
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2002 Iran Team Selection Test, 2
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2000 Taiwan National Olympiad, 3
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}_0$ by $f(1)=0$ and
\[f(n)=\max_j\{ f(j)+f(n-j)+j\}\quad\forall\, n\ge 2 \]
Determine $f(2000)$.
Gheorghe Țițeica 2025, P3
Out of all the nondegenerate triangles with positive integer sides and perimeter $100$, find the one with the smallest area.
2000 All-Russian Olympiad Regional Round, 8.7
Angle bisectors $AD$ and $CE$ of triangle $ABC$ intersect at point $O$. A line symmetrical to $ AB$ with respect to $CE$ intersects the line symmetric $BC$ with respect to $AD$, at point $K$. Prove that $KO \perp AC$.
2010 Indonesia TST, 1
Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$.
[i]Nanang Susyanto, Jogjakarta[/i]
2006 Germany Team Selection Test, 3
Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?
2022 Kosovo National Mathematical Olympiad, 3
Find all positive integers $n$ such that $10^n+3^n+2$ is a palindrome number.
2006 Bosnia and Herzegovina Team Selection Test, 1
Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?
2018 Iran MO (1st Round), 13
Bahman wants to build an area next to his garden's wall for keeping his poultry. He has three fences each of length $10$ meters. Using the garden's wall, which is straight and long, as well as the three pieces of fence, what is the largest area Bahman can enclose in meters squared?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 50+25 \sqrt 3\qquad\textbf{(C)}\ 50 + 50\sqrt 2\qquad\textbf{(D)}\ 75 \sqrt 3 \qquad\textbf{(E)}\ 300$
1998 IMO Shortlist, 1
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)
2014 Romania National Olympiad, 3
Let $ n $ be a natural number, and $ A $ the set of the first $ n $ natural numbers. Find the number of nondecreasing functions $ f:A\longrightarrow A $ that have the property
$$ x,y\in A\implies |f(x)-f(y)|\le |x-y|. $$
2017 Dutch IMO TST, 3
Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$.
Show that the smallest $k-pable$ integer is coprime to $k$.
1986 AMC 8, 14
If $ 200 \le a \le 400$ and $ 600 \le b \le 1200$, then the largest value of the quotient $ \frac{b}{a}$ is
\[ \textbf{(A)}\ \frac{3}{2} \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 300 \qquad
\textbf{(E)}\ 600 \qquad
\]
2012 CHMMC Spring, 8
A special kind of chess knight is in the origin of an infinite grid. It can make one of twelve different moves: it can move directly up, down, left, or right one unit square, or it can move $1$ units in one direction and $3$ units in an orthogonal direction. How many different squares can it be on after $2$ moves?
2013 Korea Junior Math Olympiad, 1
Compare the magnitude of the following three numbers.
$$
\sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}}
$$
2015 Tuymaada Olympiad, 7
In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$.
Find possible values of $\angle CED$
[i]D. Shiryaev [/i]