Found problems: 85335
1979 All Soviet Union Mathematical Olympiad, 270
A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$. Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area.
2009 Cono Sur Olympiad, 1
The four circles in the figure determine 10 bounded regions. $10$ numbers summing to $100$ are written in these regions, one in each region. The sum of the numbers contained in each circle is equal to $S$ (the same quantity for each of the four circles). Determine the greatest and smallest possible values of $S$.
2017 Peru IMO TST, 13
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]
1977 IMO Longlists, 24
Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation
\[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]
2020 Korea National Olympiad, 6
Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$, and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$.
Consider the set of circle $\Omega$ satisfying the following condition.
Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$.
Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$.
Prove that $AP+AQ$ is constant.
III Soros Olympiad 1996 - 97 (Russia), 10.6
There are $76$ cards with different numbers written on them. These cards are laid out on the table in a circle, number down. Try to find some three cards in a row such that the number written on the middle of these three cards is greater than on each of the two neighboring ones. You can turn over no more than $10$ cards in succession. How should one proceed to be sure to find three cardboard boxes for which the specified condition is met?
2007 Singapore Junior Math Olympiad, 2
Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.
2024 Israel National Olympiad (Gillis), P2
A positive integer $x$ satisfies the following:
\[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\]
Find all possible values of
\[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\]
where $\{y\}$ denotes the fractional part of $y$.
2017 Harvard-MIT Mathematics Tournament, 4
Find all pairs $(a,b)$ of positive integers such that $a^{2017}+b$ is a multiple of $ab$.
2021 Princeton University Math Competition, 6
Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.
2013 Harvard-MIT Mathematics Tournament, 20
The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.
2016 Iranian Geometry Olympiad, 5
Let the circles $\omega$ and $\omega'$ intersect in points $A$ and $B$. The tangent to circle $\omega$ at $A$ intersects $\omega'$ at $C$ and the tangent to circle $\omega'$ at $A$ intersects $\omega$ at $D$. Suppose that the internal bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $E$ and $F$, respectively, and the external bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $X$ and $Y$, respectively. Prove that the perpendicular bisector of $XY$ is tangent to the circumcircle of triangle $BEF$.
[i]Proposed by Mahdi Etesami Fard[/i]
2022 Rioplatense Mathematical Olympiad, 2
Four teams $A$, $B$, $C$ and $D$ play a football tournament in which each team plays exactly two times against each of the remaining three teams (there are $12$ matches). In each matchif it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $3$ points and the loser gets $0$ points.
At the end of the tournament the teams $A$, $B$ and $C$ have $8$ points each. Determine all possible points of team $D$.
2000 Saint Petersburg Mathematical Olympiad, 9.2
Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic.
[I]Proposed by S. Berlov[/i]
2017 AMC 12/AHSME, 5
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2=40$, first quartile $Q_1=33$, and third quartile $Q_3=43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3-Q_1$. How many outliers does this data set have?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
2011 Saudi Arabia Pre-TST, 4.2
Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$
2018 Singapore Junior Math Olympiad, 4
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.
2014 IPhOO, 6
A square plate has side length $L$ and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis $\overline{MN}$ where $M$ and $N$ are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is $kmL^2$, where $m$ is the mass of the plate and $k$ is a real constant. Find $k$.
[color=red]Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.[/color]
[i]Problem proposed by Ahaan Rungta[/i]
1984 Austrian-Polish Competition, 2
Let $A$ be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of $n\in A$ yields a number $f (n) \in A$ (for instance, $f (3111) = 1333$). Find those $n \in A$ with $n > f (n)$ for which $gcd(n, f (n))$ is the largest possible.
2016 Saudi Arabia IMO TST, 1
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
2011 Harvard-MIT Mathematics Tournament, 1
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.
2022 Korea Junior Math Olympiad, 4
Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that
forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.
2000 Portugal MO, 4
Calculates the sum of all numbers that can be formed using each of the odd digits once, that is, the numbers $13579$, $13597$, ..., $97531$.
2019 JHMT, 8
In $\vartriangle ABC$, $m\angle A = 90^o$, $m\angle B = 45^o$, and $m\angle C = 45^o$. Point $P$ inside $\vartriangle ABC$ satisfies $m \angle BPC =135^o$. Given that $\vartriangle PAC$ is isosceles, the largest possible value of $\tan \angle PAC$ can be expressed as $s+t\sqrt{u}$, where $s$ and $t$ are integers and $u$ is a positive integer not divisible by the square of any prime. Compute $100s + 10t + u$.
2007 Dutch Mathematical Olympiad, 3
Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$?
If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.