Found problems: 85335
1970 Dutch Mathematical Olympiad, 3
The points $P,Q,R$ and $A,B,C,D$ lie on a circle (clockwise) such that $\vartriangle PQR$ is equilateral and $ABCD$ is a square. The points $A$ and $P$ coincide. Prove that the symmetric of $B$ and $D$ wrt $PQ$ and $PR$ respectively lie on the sidelines of the symmetric square wrt $QR$.
2003 China Team Selection Test, 1
$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying:
(1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and
(2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince.
Find $|B_n^m|$ and $|B_6^3|$.
2025 Harvard-MIT Mathematics Tournament, 10
A plane $\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths $45, 66, 63, 55, 54,$ and $77,$ in that order. Compute the distance from the center of the rectangular prism to $\mathcal{P}.$
2011 Indonesia TST, 4
Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$.
[hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]
2013 AMC 10, 6
The average age of $33$ fifth-graders is $11$. The average age of $55$ of their parents is $33$. What is the average age of all of these parents and fifth-graders?
$\textbf{(A) }22\qquad\textbf{(B) }23.25\qquad\textbf{(C) }24.75\qquad\textbf{(D) }26.25\qquad\textbf{(E) }28$
2004 All-Russian Olympiad, 4
A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$
2018 China Northern MO, 4
For $n(n\geq3)$ positive intengers $a_1,a_2,\cdots,a_n$. Put the numbers on a circle. In each operation, calculate difference between two adjacent numbers and take its absolute value. Put the $n$ numbers we get on another ciecle (do not change their order). Find all $n$, satisfying that no matter how $a_1,a_2,\cdots,a_n$ are given, all numbers on the circle are equal after limited operations.
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 $BX=CY$.
2002 Swedish Mathematical Competition, 1
$268$ numbers are written around a circle. The $17$th number is $3$, the $83$rd is $4$ and the $144$th is $9$. The sum of every $20$ consecutive numbers is $72$. Find the $210$th number.
2014 ASDAN Math Tournament, 9
Find the sum of all real numbers $x$ such that $x^4-2x^3+3x^2-2x-2014=0$.
2010 JBMO Shortlist, 1
$\textbf{Problem G1}$
Consider a triangle $ABC$ with $\angle ACB=90^{\circ}$. Let $F$ be the foot of the altitude from
$C$. Circle $\omega$ touches the line segment $FB$ at point $P$, the altitude $CF$ at point $Q$ and the
circumcircle of $ABC$ at point $R$. Prove that points $A, Q, R$ are collinear and $AP = AC$.
2012 Today's Calculation Of Integral, 806
Let $n$ be positive integers and $t$ be a positive real number.
Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$
2001 Poland - Second Round, 1
Let $k,n>1$ be integers such that the number $p=2k-1$ is prime. Prove that, if the number $\binom{n}{2}-\binom{k}{2}$ is divisible by $p$, then it is divisible by $p^2$.
2001 Bundeswettbewerb Mathematik, 2
For each $ n \in \mathbb{N}$ we have two numbers $ p_n, q_n$ with the following property: For exactly $ n$ distinct integer numbers $ x$ the number \[ x^2 \plus{} p_n \cdot x \plus{} q_n\] is the square of a natural number. (Note the definition of natural numbers includes the zero here.)
PEN A Problems, 48
Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.
KoMaL A Problems 2020/2021, A. 788
Solve the following system of equations:
$$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$
2016 Taiwan TST Round 3, 2
There's a convex $3n$-polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots $A$, $B$ and $C$ form a triangle if $A$'s laser points at $B$, $B$'s laser points at $C$, and $C$'s laser points at $A$. Find the minimum number of moves that can guarantee $n$ triangles on the plane.
2017 Greece JBMO TST, 1
Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that
$$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$
Also, find the values of $a,b,c$ for which the equality happens.
2016 Switzerland Team Selection Test, Problem 10
Let $ABC$ be a non-rectangle triangle with $M$ the middle of $BC$. Let $D$ be a point on the line $AB$ such that $CA=CD$ and let $E$ be a point on the line $BC$ such that $EB=ED$. The parallel to $ED$ passing through $A$ intersects the line $MD$ at the point $I$ and the line $AM$ intersects the line $ED$ at the point $J$. Show that the points $C, I$ and $J$ are aligned.
1985 Swedish Mathematical Competition, 4
Let $p(x)$ be a polynomial of degree $n$ with real coefficients such that $p(x) \ge 0$ for all $x$. Prove that $p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0$.
2018 China Team Selection Test, 3
Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.
2017 India PRMO, 10
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?
2000 Romania Team Selection Test, 1
Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$.
[i]Mircea Becheanu [/i]
2005 Miklós Schweitzer, 10
Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.
2010 APMO, 1
Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.