Found problems: 85335
2006 Alexandru Myller, 1
Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $
2022 Middle European Mathematical Olympiad, 3
Let $ABCD$ be a parallelogram with $\angle DAB < 90$ Let $E$ be the point on the line $BC$ such that $AE = AB$ and let $F$ be the point on the line $CD$ such that $AF = AD$. The circumcircle of the triangle $CEF$ intersects the line $AE$ again in $P$ and the line $AF$ again in $Q$. Let $X$ be the reflection of $P$ over the line $DE$ and $Y$ the reflection of $Q$ over the line $BF$. Prove that $A, X, Y$ lie on the same line.
2014 All-Russian Olympiad, 3
There are $n$ cells with indices from $1$ to $n$. Originally, in each cell, there is a card with the corresponding index on it. Vasya shifts the card such that in the $i$-th cell is now a card with the number $a_i$. Petya can swap any two cards with the numbers $x$ and $y$, but he must pay $2|x-y|$ coins. Show that Petya can return all the cards to their original position, not paying more than $|a_1-1|+|a_2-2|+\ldots +|a_n-n|$ coins.
2021 Alibaba Global Math Competition, 5
Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that
(a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and
(b) the Euclidean norm of every point $v$ in $A$ satisfies
\[\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.\]
Prove that the cardinality of $A$ is at most $2d+4$.
2017 Junior Regional Olympiad - FBH, 1
Price of the book increased by $20\%$, and then decreased by $10\%$. How many percents should we decrease current price so we get a price which is $54\%$ percent of an original one?
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6
Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$. Let $ a \equal{} AB$ and $ b \equal{} CD$. For $ MN\parallel{}AB$ such that $ M$ lies on $ AD,$ $ N$ lies on $ BC$, and the trapezoids $ ABNM$ and $ MNCD$ have the same area, the length of $ MN$ equals
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number6.jpg[/img]
A. $ \sqrt{ab}$
B. $ \frac{a\plus{}b}{2}$
C. $ \frac{a^2 \plus{} b^2}{a\plus{}b}$
D. $ \sqrt{\frac{a^2 \plus{} b^2}{2}}$
E. $ \frac{a^2 \plus{} (2 \sqrt{2} \minus{} 2)ab \plus{} b^2}{\sqrt{2} (a\plus{}b)}$
2014 Kazakhstan National Olympiad, 2
$\mathbb{Q}$ is set of all rational numbers. Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that for all $x$, $y$, $z$ $\in\mathbb{Q}$ satisfy
$f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$
1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7
If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals
A. -12
B. 24
C. 35
D. -61
E. -63
2014 Brazil Team Selection Test, 4
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
2012 Singapore Senior Math Olympiad, 1
A circle $\omega$ through the incentre$ I$ of a triangle $ABC$ and tangent to $AB$ at $A$, intersects the segment $BC$ at $D$ and the extension of$ BC$ at $E$. Prove that the line $IC$ intersects $\omega$ at a point $M$ such that $MD=ME$.
2003 Romania National Olympiad, 3
For every positive integer $ n$ consider
\[ A_n\equal{}\sqrt{49n^2\plus{}0,35n}.
\]
(a) Find the first three digits after decimal point of $ A_1$.
(b) Prove that the first three digits after decimal point of $ A_n$ and $ A_1$ are the same, for every $ n$.
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).
2019 Iran MO (3rd Round), 2
Let $n,k$ be positive integers so that $n \ge k$.Find the maximum number of binary sequances of length $n$ so that fixing any arbitary $k$ bits they do not produce all binary sequances of length $k$.For exmple if $k=1$ we can only have one sequance otherwise they will differ in at least one bit which means that bit produces all binary sequances of length $1$.
2012 Regional Olympiad of Mexico Center Zone, 6
A board of $2n$ x $2n$ is colored chess style, a movement is the changing of colors of a $2$ x $2$ square. For what integers $n$ is possible to complete the board with one color using a finite number of movements?
2015 Kyiv Math Festival, P1
Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$
2015 NIMO Summer Contest, 1
For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$.
[i] Proposed by David Altizio [/i]
2015 AIME Problems, 7
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
[asy]
pair A,B,C,D,E,F,G,H,J,K,L,M,N;
B=(0,0);
real m=7*sqrt(55)/5;
J=(m,0);
C=(7*m/2,0);
A=(0,7*m/2);
D=(7*m/2,7*m/2);
E=(A+D)/2;
H=(0,2m);
N=(0,2m+3*sqrt(55)/2);
G=foot(H,E,C);
F=foot(J,E,C);
draw(A--B--C--D--cycle);
draw(C--E);
draw(G--H--J--F);
pair X=foot(N,E,C);
M=extension(N,X,A,D);
K=foot(N,H,G);
L=foot(M,H,G);
draw(K--N--M--L);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$E$",E,dir(90));
label("$F$",F,NE);
label("$G$",G,NE);
label("$H$",H,W);
label("$J$",J,S);
label("$K$",K,SE);
label("$L$",L,SE);
label("$M$",M,dir(90));
label("$N$",N,dir(180));
[/asy]
2018 Estonia Team Selection Test, 4
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
LMT Guts Rounds, 31
In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L, M,$ and $T$ such that consecutive multiples of $n,$ for any positive integer $n,$ are not assigned the same letter?
2017 CHKMO, Q1
A, B and C are three persons among a set P of n (n[u]>[/u]3) persons. It is known that A, B and C are friends of one another, and that every one of the three persons has already made friends with more than half the total number of people in P. Given that every three persons who are friends of one another form a [i]friendly group[/i], what is the minimum number of friendly groups that may exist in P?
1961 AMC 12/AHSME, 25
Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in angle $B$ is:
${{ \textbf{(A)}\ 25 \frac{5}{7} \qquad\textbf{(B)}\ 26 \frac{1}{3} \qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40}\qquad\textbf{(E)}\ \text{Not determined by the information given} } $
1991 Tournament Of Towns, (302) 3
Prove that $$\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1$$
This means
$1/(2+ (1/(3+ (1/(4+(...+1/1991))))))
+1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)$
(G. Galperin, Moscow-Tel Aviv)
2007 Federal Competition For Advanced Students, Part 2, 1
For which non-negative integers $ a<2007$ the congruence $ x^2\plus{}a \equiv 0 \mod 2007$ has got exactly two different non-negative integer solutions?
That means, that there exist exactly two different non-negative integers $ u$ and $ v$ less than $ 2007$, such that $ u^2\plus{}a$ and $ v^2\plus{}a$ are both divisible by $ 2007$.
2018 PUMaC Combinatorics A, 4
If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.
2018 Bundeswettbewerb Mathematik, 2
Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$.
a) By giving a concrete example, show that such a function exists.
b) For each such function define the sum
\[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\]
Determine all possible values of $S_f$.