Found problems: 85335
2009 Junior Balkan Team Selection Test, 4
For positive real numbers $ x,y,z$ the inequality
\[\frac1{x^2\plus{}1}\plus{}\frac1{y^2\plus{}1}\plus{}\frac1{z^2\plus{}1}\equal{}\frac12\]
holds. Prove the inequality
\[\frac1{x^3\plus{}2}\plus{}\frac1{y^3\plus{}2}\plus{}\frac1{z^3\plus{}2}<\frac13.\]
2001 AMC 10, 14
A charity sells 140 benefit tickets for a total of $ \$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
$ \textbf{(A)} \ \$782 \qquad \textbf{(B)} \ \$986 \qquad \textbf{(C)} \ \$1158 \qquad \textbf{(D)} \ \$1219 \qquad \textbf{(E)} \ \$1449$
2003 JHMMC 8, 13
A problem author for a math competition was looking through a tentative exam when he realized that
he could not use one of his proposed problems. Frustrated, he decided to take a nap instead, and slept
from $10:47\text{ AM}$ to $7:32\text{ PM}$. For how many minutes did he sleep?
2010 IFYM, Sozopol, 7
Let $\Delta ABC$ be an isosceles triangle with base $AB$. Point $P\in AB$ is such that $AP=2PB$. Point $Q$ from the segment $CP$ is such that $\angle AQP=\angle ACB$. Prove that $\angle PQB=\frac{1}{2}\angle ACB$.
2011 Costa Rica - Final Round, 1
Let $ABC$ be a triangle with orthocenter $H$. Let $P,Q,R$ be the reflections of $H$ with respect to sides $BC,AC,AB$, respectively. Show that $H$ is incenter of $PQR$.
2020 Yasinsky Geometry Olympiad, 3
The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively.
(Grigory Filippovsky)
1959 Miklós Schweitzer, 7
[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series
$\sum_{n=1}^{\infty} \epsilon_n z_n$
is convergente. [b](F. 9)[/b]
2009 Saint Petersburg Mathematical Olympiad, 3
Streets of Moscow are some circles (rings) with common center $O$ and some straight lines from center $O$ to external ring. Point $A,B$ - two crossroads on external ring. Three friends want to move from $A$ to $B$. Dima goes by external ring, Kostya goes from $A$ to $O$ then to $B$. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.
2002 USAMTS Problems, 1
The sequence of letters [b]TAGC[/b] is written in succession 55 times on a strip, as shown below. The strip is to be cut into segments between letters, leaving strings of letters on each segment, which we call words. For example, a cut after the first G, after the second T, and after the second C would yield the words [b]TAG[/b], [b]CT[/b] and [b]AGC[/b]. At most how many distinct words could be found if the entire strip were cut? Justify your answer.
\[\boxed{\textbf{T A G C T A G C T A G}}\ldots\boxed{\textbf{C T A G C}}\]
2017 Czech-Polish-Slovak Match, 1
Let ${ABC}$ be a triangle. Line [i]l[/i] is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on [i]l[/i]. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear.
(Slovakia)
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.