Found problems: 85335
1990 Mexico National Olympiad, 3
Show that $n^{n-1}-1$ is divisible by$ (n-1)^2$ for $n > 2$.
2019 New Zealand MO, 5
Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.
1993 Tournament Of Towns, (374) 2
A square is constructed on the side $AB$ of triangle $ABC$ (outside the triangle).$ O$ is the centre of the square. $M$ and $N$ are the midpoints of the sides $BC$ and $AC$. The lengths of these sides are $a$ and $b$ respectively. Find the maximal possible value of the sum $CM + ON$ (when the angle at $C$ changes).
(IF Sharygin)
2011 India National Olympiad, 3
Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$
2024 Ukraine National Mathematical Olympiad, Problem 6
You are given a convex hexagon with parallel opposite sides. For each pair of opposite sides, a line is drawn parallel to these sides and equidistant from them. Prove that the three lines thus obtained intersect at one point if and only if the lengths of the opposite sides are equal.
[i]Proposed by Nazar Serdyuk[/i]
2013 Saudi Arabia BMO TST, 5
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.
2022 Thailand Mathematical Olympiad, 1
Let $BC$ be a chord of a circle $\Gamma$ and $A$ be a point inside $\Gamma$ such that $\angle BAC$ is acute. Outside $\triangle ABC$, construct two isosceles triangles $\triangle ACP$ and $\triangle ABR$ such that $\angle ACP$ and $\angle ABR$ are right angles. Let lines $BA$ and $CA$ meet $\Gamma$ again at points $E$ and $F$, respectively. Let lines $EP$ and $FR$ meet $\Gamma$ again at points $X$ and $Y$, respectively. Prove that $BX=CY$.
2024 CCA Math Bonanza, L3.1
Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability the sum of the numbers the die landed on is divisible by $7$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Lightning 3.1[/i]
2006 Bundeswettbewerb Mathematik, 2
Find all functions $f: Q^{+}\rightarrow R$ such that
$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$
2017 Iran MO (3rd round), 3
Let $ABC$ be an acute-angle triangle. Suppose that $M$ be the midpoint of $BC$ and $H$ be the orthocenter of $ABC$. Let $F\equiv BH\cap AC$ and $E\equiv CH\cap AB$. Suppose that $X$ be a point on $EF$ such that $\angle XMH=\angle HAM$ and $A,X$ are in the distinct side of $MH$. Prove that $AH$ bisects $MX$.
2019 AMC 10, 8
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?
[asy]
pen white = gray(1);
pen gray = gray(0.5);
draw((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle);
fill((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle, gray);
draw((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle);
fill((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle, white);
draw((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle);
fill((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle, white);
draw((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle);
fill((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle, white);
draw((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle);
fill((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle, white);
[/asy]
$\textbf{(A) } 4\qquad\textbf{(B) }12 - 4\sqrt{3} \qquad\textbf{(C) } 3\sqrt{3} \qquad \textbf{(D) }4\sqrt{3}\qquad \textbf{(E) }16 - \sqrt{3}$
1981 Austrian-Polish Competition, 2
The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.
2013 ELMO Shortlist, 5
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
2016 PUMaC Number Theory A, 7
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)
2009 Rioplatense Mathematical Olympiad, Level 3, 1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f(xy)=\max\{f(x+y),f(x) f(y)\} \]
for all real numbers $x$ and $y$.
2023 Junior Balkan Team Selection Tests - Romania, P4
Let $ABC$ be an acute triangle with $\angle B > \angle C$. On the circle $\mathcal{C}(O, R)$ circumscribed to this triangle points $D, E, J, K, S$ are chosen such that $A, E, J$ and $K$ are on the same side of the line $BC$, the diameter $DE$ is perpendicular on the chord $BC$, $S\in \overarc{EK},\overarc{AE}=\overarc{BJ}=\overarc{CK}=\dfrac{1}{4}\overarc{CE}$ . Let $\{F\}=AC\cap DE, \{M\}=BK\cap AD, \{P\}=BK\cap AC$ and $\{Q\}=CJ\cap BF$. If $\angle SMK =30^{\circ}$ and $\angle AQP = 90^{\circ}$, show that the line $MS$ is tangent to the circumscribed circle of triangle $AOF$.
2016 Abels Math Contest (Norwegian MO) Final, 4
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \]
Holds for all $x \not= y \in \mathbb{R}$
2015 ASDAN Math Tournament, 8
Let $\{x\}$ denote the fractional part of $x$, which means the unique real $0\leq\{x\}<1$ such that $x-\{x\}$ is an integer. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $[m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and $b$ range along all real numbers.
2003 National High School Mathematics League, 7
The solution set for inequality $|x|^3-2x^2-4|x|+3<0$ is________.
2018 Sharygin Geometry Olympiad, 10
In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?
2016 NIMO Problems, 1
Three fair six-sided dice are labeled with the numbers $\{1, 2, 3, 4, 5, 6\},$ $\{1, 2, 3, 4, 5, 6\},$ and $\{1, 2, 3, 7, 8, 9\},$ respectively. All three dice are rolled. The probability that at least two of the dice have the same value is $m/n,$ where $m, n$ are relatively prime positive integers. Find $100m + n.$
[i]Proposed by Michael Tang[/i]
2009 AMC 12/AHSME, 12
How many positive integers less than $ 1000$ are $ 6$ times the sum of their digits?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 12$
2013 Brazil Team Selection Test, 2
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 $n$?
2000 Tuymaada Olympiad, 3
Polynomial $ P(t)$ is such that for all real $ x$,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?
1961 All-Soviet Union Olympiad, 4
Given are arbitrary integers $a,b,p$. Prove that there always exist relatively prime integers $k$ and $\ell$ such that $ak+b\ell$ is divisible by $p$.