Found problems: 85335
2020 Online Math Open Problems, 25
Let $n$ be a positive integer with exactly twelve positive divisors $1=d_1 < \cdots < d_{12}=n$. We say $n$ is [i]trite[/i] if \[
5 + d_6(d_6+d_4) = d_7d_4.
\] Compute the sum of the two smallest trite positive integers.
[i]Proposed by Brandon Wang[/i]
2009 China Team Selection Test, 3
Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$
2018-2019 Fall SDPC, 7
The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.
2022 Belarusian National Olympiad, 9.1
Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$.
Prove that points $X,Y,Z$ and $W$ lie on a circle.
2012 IMO Shortlist, C2
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
2013 NIMO Problems, 7
Tyler has two calculators, both of which initially display zero. The first calculators has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.
A positive integer $n$ is called [i]ambivalent[/i] if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
[i]Proposed by Joshua Xiong[/i]
2005 MOP Homework, 1
Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).
2024 Israel Olympic Revenge, P2
Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.
2000 Poland - Second Round, 4
Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.
1954 AMC 12/AHSME, 40
If $ \left (a\plus{}\frac{1}{a} \right )^2\equal{}3$, then $ a^3\plus{}\frac{1}{a^3}$ equals:
$ \textbf{(A)}\ \frac{10\sqrt{3}}{3} \qquad
\textbf{(B)}\ 3\sqrt{3} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 7\sqrt{7} \qquad
\textbf{(E)}\ 6\sqrt{3}$
2018 Romanian Masters in Mathematics, 2
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
PEN A Problems, 67
Prove that $2n \choose n$ is divisible by $n+1$.
2022 Singapore MO Open, Q5
Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer.
[i]Proposed by fattypiggy123[/i]
2023 SG Originals, Q4
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
2016 Moldova Team Selection Test, 7
Let $\Omega$ and $O$ be the circumcircle of acute triangle $ABC$ and its center, respectively. $M\ne O$ is an arbitrary point in the interior of $ABC$ such that $AM$, $BM$, and $CM$ intersect $\Omega$ at $A_{1}$, $B_{1}$, and $C_{1}$, respectiuvely. Let $A_{2}$, $B_{2}$, and $C_{2}$ be the circumcenters of $MBC$, $MCA$, and $MAB$, respectively. It is to be proven that $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C{2}$ concur.
2012 IFYM, Sozopol, 4
In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality:
$\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.
Kvant 2022, M2697
There are some gas stations on a circular highway. The total amount of gasoline in them is enough for two laps. Two drivers want to refuel at one station and starting from it, go in different directions, both of them completing an entire lap. Along the way, they can refuel at other stations, without necessarily taking all the gasoline. Prove that drivers will always be able to do this.
[i]Proposed by I. Bogdanov[/i]
2015 IFYM, Sozopol, 3
Let $ a,b,c>0$ prove that:\[
\frac{a^{3}}{(a+b)^{3}}+\frac{b^{3}}{(b+c)^{3}}+\frac{c^{3}}{(c+a)^{3}}\geq
\frac{3}{8} \]
Good luck! :D
2009 Sharygin Geometry Olympiad, 8
Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square?
(C.Pohoata, A.Zaslavsky)
1935 Moscow Mathematical Olympiad, 017
Solve the system $\begin{cases} x^3 - y^3 = 26 \\
x^2y - xy^2 = 6
\end{cases}$ in $C$
[hide=other version]solved below
Solve the system $\begin{cases} x^3 - y^3 = 2b \\
x^2y - xy^2 = b
\end{cases}$[/hide]
2016 IFYM, Sozopol, 2
A cell is cut from a chessboard $8\, x\, 8$, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length $\sqrt{17}$ or $\sqrt{65}$. When is the number of such broken lines bigger – when the cut cell is $(1,2)$ or $(3,6)$? (The rows and columns on the board are numerated consecutively from 1 to 8.)
2016 Indonesia MO, 8
Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.
2012 China Northern MO, 4
There are $n$ ($n \ge 4$) straight lines on the plane. For two straight lines $a$ and $b$, if there are at least two straight lines among the remaining $n-2$ lines that intersect both straight lines $a$ and $b$, then $a$ and $b$ are called a [i]congruent [/i] pair of staight lines, otherwise it is called a [i]separated[/i] pair of straight lines. If the number of [i]congruent [/i] pairs of straight line among $n$ straight lines is $2012$ more than the number of [i]separated[/i] pairs of straight line , find the smallest possible value of $n$ (the order of the two straight lines in a pair is not counted).
1967 IMO Longlists, 32
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
1976 IMO Longlists, 7
Let $P$ be a fixed point and $T$ a given triangle that contains the point $P$. Translate the triangle $T$ by a given vector $\bold{v}$ and denote by $T'$ this new triangle. Let $r, R$, respectively, be the radii of the smallest disks centered at $P$ that contain the triangles $T , T'$, respectively. Prove that $r + |\bold{v}| \leq 3R$ and find an example to show that equality can occur.