Found problems: 85335
2024 Centroamerican and Caribbean Math Olympiad, 5
Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations:
\[
\begin{cases}
\sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\
\sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2
\end{cases}
\]
Find the maximum value of \(x + y\).
2011 Purple Comet Problems, 17
In how many distinguishable rearrangements of the letters ABCCDEEF does the A precede both C's, the F appears between the 2 C's, and the D appears after the F?
2014 Korea Junior Math Olympiad, 1
Given $\triangle ABC$ with incenter $I$. Line $AI$ meets $BC$ at $D$. The incenter of $\triangle ABD, \triangle ADC$ are $E,F$, respectively. Line $DE$ meets the circumcircle of $\triangle BCE$ at$ P(\neq E)$ and line $DF$ meets the circumcircle of $\triangle BCF$ at$ Q(\neq F)$.
Show that the midpoint of $BC$ lies on the circumcircle of $\triangle DPQ$.
Russian TST 2016, P2
$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.
1990 IMO Longlists, 24
Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$:
\[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]
1981 IMO Shortlist, 12
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
MathLinks Contest 1st, 3
Let $x_0 = 1$ and $x_1 = 2003$ and define the sequence $(x_n)_{n \ge 0}$ by: $x_{n+1} =\frac{x^2_n + 1}{x_{n-1}}$ , $\forall n \ge 1$
Prove that for every $n \ge 2$ the denominator of the fraction $x_n$, when $x_n$ is expressed in lowest terms is a power of $2003$.
2007 Iran MO (3rd Round), 2
Let $ m,n$ be two integers such that $ \varphi(m) \equal{}\varphi(n) \equal{} c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.
2016 SEEMOUS, Problem 2
SEEMOUS 2016 COMPETITION PROBLEMS
1950 Polish MO Finals, 2
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.
Russian TST 2020, P3
In a convex quadrilateral $ABCD$, the lines $AB$ and $DC$ intersect at point $P{}$ and the lines $AD$ and $BC$ intersect at point $Q{}$. The points $E{}$ and $F{}$ are inside the quadrilateral $ABCD$ such that the circles $(ABE), (CDE), (BCF),(ADF)$ intersect at one point $K{}$. Prove that the circles $(PKF)$ and $(QKE)$ intersect a second time on the line $PQ$.
1989 APMO, 5
Determine all functions $f$ from the reals to the reals for which
(1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$,
where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)
2013 NIMO Summer Contest, 9
Compute $99(99^2+3) + 3\cdot99^2$.
[i]Proposed by Evan Chen[/i]
2001 AMC 12/AHSME, 24
In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$.
[asy]
pair A, B, C, D;
A = origin;
real Bcoord = 3*sqrt(2) + sqrt(6);
B = Bcoord/2*dir(180);
C = sqrt(6)*dir(120);
draw(A--B--C--cycle);
D = (C-B)/2.4 + B;
draw(A--D);
label("$A$", A, dir(0));
label("$B$", B, dir(180));
label("$C$", C, dir(110));
label("$D$", D, dir(130));
[/asy]
$ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$
2015 Turkey MO (2nd round), 1
$m$ and $n$ are positive integers. If the number \[ k=\dfrac{(m+n)^2}{4m(m-n)^2+4}\] is an integer, prove that $k$ is a perfect square.
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.
1976 IMO Longlists, 16
Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.
1995 IMO Shortlist, 2
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that
\[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]
1996 Estonia National Olympiad, 5
Three children wanted to make a table-game. For that purpose they wished to enumerate the $mn$ squares of an $m \times n$ game-board by the numbers $1, ... ,mn$ in such way that the numbers $1$ and $mn$ lie in the corners of the board and the squares with successive numbers have a common edge. The children agreed to place the initial square (with number $1$) in one of the corners but each child wanted to have the final square (with number $mn$ ) in different corner. For which numbers $m$ and $n$ is it possible to satisfy the wish of any of the children?
2005 MOP Homework, 5
Show that for nonnegative integers $m$ and $n$,
$\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$
$=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.
2022 Argentina National Olympiad, 1
For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..
2024 Nepal Mathematics Olympiad (Pre-TST), Problem 4
Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer.
[i]Proposed by Prajit Adhikari, Nepal[/i]
2021 BMT, 2
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?
1988 IMO Longlists, 10
Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$
1989 IMO Shortlist, 12
There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.