Found problems: 85335
2005 Kazakhstan National Olympiad, 1
Solve equation
\[2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|\]
2021 Taiwan TST Round 1, N
For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers.
[i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.
2018 Middle European Mathematical Olympiad, 2
The two figures depicted below consisting of $6$ and $10$ unit squares, respectively, are called staircases.
Consider a $2018\times 2018$ board consisting of $2018^2$ cells, each being a unit square. Two arbitrary
cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated).
2023 MOAA, Tie
TB1. Two not necessarily distinct positive integers $a,b$ are randomly chosen from the set $\{1,2,\ldots, 20\}$. Find the expected value of the number of distinct prime factors of $ab$.
[i]Proposed by Harry Kim[/i]
TB2. Square $ABCD$ has side length $15$. Let $E$ and $F$ be points on $AD$ and $BC$ respectively such that $AE = 5$ and $BF = 5$. Find the area of intersection between triangles $\triangle{AFC}$ and $\triangle{BED}$.
[i]Proposed by Andy Xu[/i]
TB3. If $x$ and $y$ satisfy $$\frac{1}{x}+\frac{1}{y} = 2$$ $$\frac{x}{y}+\frac{y}{x} = 3$$ find $xy$.
[i]Proposed by Harry Kim and Andy Xu[/i]
1981 All Soviet Union Mathematical Olympiad, 304
Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides.
2013 Macedonia National Olympiad, 2
$ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ($ n $ is natural number)
2002 Austrian-Polish Competition, 7
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$.
1948 Putnam, A3
Let $(a_n)$ be a decreasing sequence of positive numbers with limit $0$ such that
$$b_n = a_n -2 a_{n+1}+a_{n+2} \geq 0$$
for all $n.$ Prove that
$$\sum_{n=1}^{\infty} n b_n =a_1.$$
2024 ELMO Shortlist, A3
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$
[i]Andrew Carratu[/i]
2013 Saint Petersburg Mathematical Olympiad, 3
Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$.
S. Berlov
2009 ITAMO, 3
A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.
2004 Regional Competition For Advanced Students, 4
The sequence $ < x_n >$ is defined through:
$ x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1$ for $ n > 0$
Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers.
Show that there exist infinitely many prime numbers in this sequence.
2013 Dutch IMO TST, 4
Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.)
What is the minimal value of $N$ for which this is possible?
2019 ASDAN Math Tournament, 7
Consider a triangle $\vartriangle ABC$ with $AB = 7$, $BC = 8$, $CA = 9$, and area $12\sqrt5$. We draw squares on each sides, namely $BCD_2D_1$, $CAE_2E_1$ and $ABF_2F_1$, so that the interiors of the squares do not intersect the interior of the triangle. What is the area of $\vartriangle D_2E_2F_2$?
2010 LMT, 10
A two digit prime number is such that the sum of its digits is $13.$ Determine the integer.
1996 Tournament Of Towns, (521) 4
Prove that for any function $f(x)$, continuous or otherwise, $$f(f(x)) = x^2 - 1996$$ cannot hold for all real numbers $x$.
(S Bogatiy, M Smurov,)
2019 Jozsef Wildt International Math Competition, W. 54
Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$
2023 APMO, 2
Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.
1994 AMC 12/AHSME, 13
In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$
[asy]
draw((0,0)--(8,0)--(4,12)--cycle);
draw((8,0)--(1.6,4.8));
label("A", (4,12), N);
label("B", (0,0), W);
label("C", (8,0), E);
label("P", (1.6,4.8), NW);
dot((0,0));
dot((4,12));
dot((8,0));
dot((1.6,4.8));
[/asy]
$ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} $
2024 Iranian Geometry Olympiad, 1
Reflect each of the shapes $A,B$ over some lines $l_A,l_B$ respectively and rotate the shape $C$ such that a $4 \times 4$ square is obtained. Identify the lines $l_A,l_B$ and the center of the rotation, and also draw the transformed versions of $A,B$ and $C$ under these operations.
[img]https://s8.uupload.ir/files/photo14908574605_i39w.jpg[/img]
[i]Proposed by Mahdi Etesamifard - Iran[/i]
2014 District Olympiad, 4
Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with
the properties:
[list=a]
[*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $
[*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]
1996 German National Olympiad, 5
Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.
IV Soros Olympiad 1997 - 98 (Russia), 9.3
Through point $O$ - the center of a circle circumscribed around an acute triangle - a straight line is drawn, perpendicular to one of its sides and intersecting the other two sides of the triangle (or their extensions) at points $M $ and $N$. Prove that $OM+ON \ge R$, where $R$ is the radius of the circumscribed circle around the triangle.
2002 AMC 12/AHSME, 13
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\]
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2021 CHMMC Winter (2021-22), 4
How many ordered triples $(a, b, c)$ of integers $1 \le a, b, c \le 31$ are there such that the remainder of $ab+bc+ca$ divided by $31$ equals $8$?