Found problems: 85335
2011 LMT, 4
What is the sum of the first $2011$ integers closest in value to $0,$ including $0$ itself?
2015 Baltic Way, 11
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.
2006 MOP Homework, 2
Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.
2020 Yasinsky Geometry Olympiad, 1
In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.
2017 HMNT, 2
Determine the sum of all distinct real values of $x$ such that $||| \cdots ||x|+x| \cdots |+x|+x|=1$ where there are $2017$ $x$s in the equation.
2022 Turkey EGMO TST, 6
Let $x,y,z$ be positive real numbers satisfying the equations
$$xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0$$
What is the minimum value of the ratio of the sum of the largest and smallest numbers among $x,y,z$ to the median of them.
2018 Auckland Mathematical Olympiad, 5
There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$?
2019 PUMaC Individual Finals A, B, A2
Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$,
$p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$
2013 Dutch BxMO/EGMO TST, 3
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
1993 Irish Math Olympiad, 5
For a complex number $ z\equal{}x\plus{}iy$ we denote by $ P(z)$ the corresponding point $ (x,y)$ in the plane. Suppose $ z_1,z_2,z_3,z_4,z_5,\alpha$ are nonzero complex numbers such that:
$ (i)$ $ P(z_1),...,P(z_5)$ are vertices of a complex pentagon $ Q$ containing the origin $ O$ in its interior, and
$ (ii)$ $ P(\alpha z_1),...,P(\alpha z_5)$ are all inside $ Q$.
If $ \alpha\equal{}p\plus{}iq$ $ (p,q \in \mathbb{R})$, prove that $ p^2\plus{}q^2 \le 1$ and $ p\plus{}q \tan \frac{\pi}{5} \le 1$.
1986 IMO Shortlist, 18
Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.
2021 Girls in Mathematics Tournament, 4
Mariana plays with an $8\times 8$ board with all its squares blank. She says that two houses are [i]neighbors [/i] if they have a common side or vertex, that is, two houses can be neighbors vertically, horizontally or diagonally. The game consists of filling the $64$ squares on the board, one after the other, each with a number according to the following rule: she always chooses a house blank and fill it with an integer equal to the number of neighboring houses that are still in White. Once this is done, the house is no longer considered blank.
Show that the value of the sum of all $64$ numbers written on the board at the end of the game does not depend in the order of filling. Also, calculate the value of this sum.
Note: A house is not neighbor to itself.
[hide=original wording]Mariana brinca com um tabuleiro 8 x 8 com todas as suas casas em branco. Ela diz que duas
casas s˜ao vizinhas se elas possu´ırem um lado ou um v´ertice em comum, ou seja, duas casas podem ser vizinhas
verticalmente, horizontalmente ou diagonalmente. A brincadeira consiste em preencher as 64 casas do tabuleiro,
uma ap´os a outra, cada uma com um n´umero de acordo com a seguinte regra: ela escolhe sempre uma casa
em branco e a preenche com o n´umero inteiro igual `a quantidade de casas vizinhas desta que ainda estejam em
branco. Feito isso, a casa n˜ao ´e mais considerada em branco.
Demonstre que o valor da soma de todos os 64 n´umeros escritos no tabuleiro ao final da brincadeira n˜ao depende
da ordem do preenchimento. Al´em disso, calcule o valor dessa soma.
Observa¸c˜ao: Uma casa n˜ao ´e vizinha a si mesma[/hide]
1999 Moldova Team Selection Test, 5
Let $a_1, a_2, \ldots, a_n$ be real numbers, but not all of them null. Show that the equation $$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$ has at most one real solution.
2005 Regional Competition For Advanced Students, 1
Show for all integers $ n \ge 2005$ the following chaine of inequalities:
$ (n\plus{}830)^{2005}<n(n\plus{}1)\dots(n\plus{}2004)<(n\plus{}1002)^{2005}$
2022 JHMT HS, 10
The maximum value of
\[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \]
over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.
2024 Stars of Mathematics, P3
Fix postive integer $n\geq 2$. Let $a_1,a_2,\dots ,a_n$ be real numbers in the interval $[1,2024]$. Prove that $$\sum_{i=1}^n\frac{1}{a_i}(a_1+a_2+\dots +a_i)>\frac{1}{44}n(n+33).$$
[i]Proposed by Radu-Andrei Lecoiu[/i]
2018 ASDAN Math Tournament, 6
Square $ABCD$ has side length $5$. Draw E on $BC$ and $F$ on $AD$ such that $BE < AF$. Next, flip $ABCD$ across $EF$ to a square $A'B'C'D'$ such that $C'$ lies in the interior of $ABCD$ and $C$ lies in the interior of $A'B'C'D'$. Suppose that $CC' = 4$ and $DD' = 2$. Compute $AA'$.
2022 Korea Winter Program Practice Test, 5
Let $ABDC$ be a cyclic quadrilateral inscribed in a circle $\Omega$. $AD$ meets $BC$ at $P$, and $\Omega$ meets lines passing $A$ and parallel to $DB$, $DC$ at $E$, $F$, respectively. $X$ is a point on $\Omega$ such that $PA=PX$. Prove that the lines $BE$, $CF$, and $DX$ are concurrent.
2016 ASDAN Math Tournament, 5
In the following diagram, the square and pentagon are both regular and share segment $AB$ as designated. What is the measure of $\angle CBD$ in degrees?
2013 Hitotsubashi University Entrance Examination, 3
Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$.
(1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$.
(2) If $q=p+1$, then find the minimum value of $S$.
(3) If $pq=-1$, then find the minimum value of $S$.
2010 Purple Comet Problems, 5
If $a$ and $b$ are positive integers such that $a \cdot b = 2400,$ find the least possible value of $a + b.$
Bangladesh Mathematical Olympiad 2020 Final, #4
Once in a restaurant [b][i]Dr. Strange[/i][/b] found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?
2015 All-Russian Olympiad, 3
Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$.
2016 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
2005 China Northern MO, 5
Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.