Found problems: 85335
1999 Spain Mathematical Olympiad, 5
The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that:
a) $g_a,g_b,g_c \ge \frac{2}{3}r$
b) $g_a+g_b+g_c \ge 3r$
2022 Mexican Girls' Contest, 7
Let $ABCD$ be a parallelogram (non-rectangle) and $\Gamma$ is the circumcircle of $\triangle ABD$. The points $E$ and $F$ are the intersections of the lines $BC$ and $DC$ with $\Gamma$ respectively. Define $P=ED\cap BA$, $Q=FB\cap DA$ and $R=PQ\cap CA$. Prove that
$$\frac{PR}{RQ}=(\frac{BC}{CD})^2$$
2024 LMT Fall, 1
Find the value of \[(2+0+2+4)+\left(2^0+2^4\right)+\left(2^{0^{2^4}}\right).\]
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]