Found problems: 85335
2019 Belarusian National Olympiad, 11.3
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.
Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$.
[i](I. Gorodnin)[/i]
2007 ITest, 45
Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$.
1954 AMC 12/AHSME, 43
The hypotenuse of a right triangle is $ 10$ inches and the radius of the inscribed circle is $ 1$ inch. The perimeter of the triangle in inches is:
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 22 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 26 \qquad
\textbf{(E)}\ 30$
1988 IMO Longlists, 56
Given a set of 1988 points in the plane. No four points of the set are collinear. The points of a subset with 1788 points are coloured blue, the remaining 200 are coloured red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100 red points.
1998 National Olympiad First Round, 8
$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ?
$\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$
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]