Found problems: 85335
2017 Sharygin Geometry Olympiad, P23
Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.
1995 India National Olympiad, 2
Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.
2021 Indonesia TST, N
For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.
2003 Miklós Schweitzer, 4
Let $\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}$ integers such that $a_{n,i}\neq a_{n,j}$ for $1\le i<j\le n\, , n=2,3,\ldots$ and let $\left\langle y\right\rangle\in [0,1)$ denote the fractional part of the real number $y$. Show that there exists a real sequence $\{ x_n\}_{n=1}^{\infty}$ such that the numbers $\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle$ are asymptotically uniformly distributed on the interval $[0,1]$.
(translated by L. Erdős)
1970 Miklós Schweitzer, 4
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\]
J. Suranyi
1993 All-Russian Olympiad Regional Round, 11.6
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
2020 Brazil National Olympiad, 6
Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer .
(a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$.
(b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.
2018 Czech and Slovak Olympiad III A, 1
In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. [i](Since this was the 67th year of CZE/SVK MO,)[/i] show that if the group is 6-great, then it is 7-great as well.
[b]Bonus[/b] (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is $(k+1)$-great as well.
2009 District Olympiad, 3
Let $a$ and $b$ be non-negative integers. Prove that the number $a^2 + b^2$ is the difference of two perfect squares if and only if $ab$ is even.
2022 Balkan MO Shortlist, A2
Let $k > 1{}$ be a real number, $n\geqslant 3$ be an integer, and $x_1 \geqslant x_2\geqslant\cdots\geqslant x_n$ be positive real numbers. Prove that \[\frac{x_1+kx_2}{x_2+x_3}+\frac{x_2+kx_3}{x_3+x_4}+\cdots+\frac{x_n+kx_1}{x_1+x_2}\geqslant\frac{n(k+1)}{2}.\][i]Ilija Jovcheski[/i]
2008 F = Ma, 25
Two satellites are launched at a distance $R$ from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed $v_\text{0}$ and enters a circular orbit. The second satellite, however, is launched at a speed $\frac{1}{2}v_\text{0}$. What is the minimum distance between the second satellite and the planet over the course of its orbit?
(a) $\frac{1}{\sqrt{2}}R$
(b) $\frac{1}{2}R$
(c) $\frac{1}{3}R$
(d) $\frac{1}{4}R$
(e) $\frac{1}{7}R$
1990 Baltic Way, 3
Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by
\[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\]
Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?
1986 IMO Shortlist, 21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
2022 CCA Math Bonanza, L3.3
Determine the sum of all positive integers $n<100$ satisfying the following expression. \[\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).\] Here, $\textstyle\sum$ and $\textstyle\prod$ represent sum and product, respectively.
[i]2022 CCA Math Bonanza Lightning Round 3.3[/i]
2021 IMC, 3
We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each . Find
$\text{sup}\{d| d \text{is good}\}$.
1998 Iran MO (3rd Round), 1
A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.
2018 Regional Olympiad of Mexico Southeast, 1
Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.
2001 Junior Balkan Team Selection Tests - Romania, 2
Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.
1982 IMO Shortlist, 2
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
\[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\]
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$
2021 China Second Round A1, 2
Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.
1968 AMC 12/AHSME, 25
Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is:
$\textbf{(A)}\ xy \qquad\textbf{(B)}\ \frac{y}{x+y} \qquad\textbf{(C)}\ \frac{xy}{x-1} \qquad\textbf{(D)}\ \frac{x+y}{x+1} \qquad\textbf{(E)}\ \frac{x+y}{x-1}$
2015 Federal Competition For Advanced Students, 4
A [i]police emergency number[/i] is a positive integer that ends with the digits $133$ in decimal representation. Prove that every police emergency number has a prime factor larger than $7$.
(In Austria, $133$ is the emergency number of the police.)
(Robert Geretschläger)
2022 JHMT HS, 8
In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.
2018 District Olympiad, 1
Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.
2014 Baltic Way, 12
Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$