Found problems: 85335
2011 Korea - Final Round, 3
There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'.
(i) All integers contained in $R$ are bigger than $h$.
(ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$.
Assume that all integers are given to make shelves as much as possible. Find the number of shelves.
1955 Polish MO Finals, 5
In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.
2019 Dutch IMO TST, 3
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
2018 Azerbaijan IMO TST, 2
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
2023 MIG, 3
A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle?
$\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$
2003 Junior Balkan Team Selection Tests - Moldova, 4
Let $m$ and $n$ be arbitrary positive integers, and
$a, b, c$ be different natural numbers of the form $2^m.5^n$. Determine
the number of all equations of the form $ax^2-2bx+c=0$ if it is known that each equation
has only one real solution.
2011 Sharygin Geometry Olympiad, 20
Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$.
[i]Nikolai Beluhov and Aleksey Zaslavsky[/i]
1993 IMO Shortlist, 3
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
2023 USA TSTST, 9
For every integer $m\ge 1$, let $\mathbb{Z}/m\mathbb{Z}$ denote the set of integers modulo $m$. Let $p$ be a fixed prime and let $a\ge 2$ and $e\ge 1$ be fixed integers. Given a function $f\colon \mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k\ge 0$, the $k$[i]th finite difference[/i], denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by
\begin{align*}
\Delta^0 f(n)&=f(n)\\
\Delta^k f(n)&=\Delta^{k-1}f(n+1)-\Delta^{k-1}f(n) & \text{for } k=1,2,\dots.
\end{align*}
Determine the number of functions $f$ such that there exists some $k\ge 1$ for which $\Delta^kf=f$.
[i]Holden Mui[/i]
1949-56 Chisinau City MO, 6
Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.
2010 IFYM, Sozopol, 3
Through vertex $C$ of $\Delta ABC$ are constructed lines $l_1$ and $l_2$ which are symmetrical about the angle bisector $CL_c$. Prove that the projections of $A$ and $B$ on lines $l_1$ and $l_2$ lie on one circle.
2008 Teodor Topan, 2
Let $ \sigma \in S_n$ and $ \alpha <2$. Evaluate$ \displaystyle\lim_{n\to\infty} \displaystyle\sum_{k\equal{}1}^{n}\frac{\sigma (k)}{k^{\alpha}}$.
2019 India Regional Mathematical Olympiad, 1
Suppose $x$ is a non zero real number such that both $x^5$ and $20x+\frac{19}{x}$ are rational numbers. Prove that $x$ is a rational number.
2015 Indonesia MO Shortlist, C6
Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.
2021 Balkan MO Shortlist, C1
Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple.
Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.
Denmark (Mohr) - geometry, 2013.5
The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$.
[img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]
2010 Peru MO (ONEM), 3
Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.
2018 Taiwan APMO Preliminary, 4
If we fill $1\sim 16$ into $4\times4$ chessboard randomly. What is the possibility of the sum of each rows and columns are all even?
2003 AMC 12-AHSME, 24
If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$?
$ \textbf{(A)}\ \minus{}2 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2016 AMC 10, 6
Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$
1999 Austrian-Polish Competition, 9
A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold:
(i) The endpoints of each selected segment are lattice points;
(ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$,
(iii) Each selected segment contains exactly five lattice points, all of which are selected,
(iv) Every two selected segments have at most one common point.
A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.
1970 IMO Longlists, 55
A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.
2019 South Africa National Olympiad, 1
Determine all positive integers $a$ for which $a^a$ is divisible by $20^{19}$.
2022 USAMTS Problems, 3
A positive integer $N$ is called [i]googolicious[/i] if there are exactly $10^{100}$ positive integers $x$ that satisfy \[\left\lfloor \frac{N}{\left\lfloor \frac{N}{x} \right\rfloor } \right\rfloor = x,\] where $z$ denotes the greatest integer less than $z.$ Find, with proof, all googolicious integers $N.$
2012 Math Prize For Girls Problems, 1
In the morning, Esther biked from home to school at an average speed of $x$ miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of $x$?