Found problems: 85335
2014 ELMO Shortlist, 10
We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear.
[i]Proposed by Sammy Luo[/i]
1958 Miklós Schweitzer, 9
[b]9.[/b] Show that if $f(z) = 1+a_1 z+a_2z^2+\dots$ for $\mid z \mid\leq 1$ and
$\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2$,
then $f(z)$ has a root in the disc $\mid z \mid \leq 1$.[b](F. 4)[/b]
2018 HMNT, 3
A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].
1998 Switzerland Team Selection Test, 2
Find all nonnegative integer solutions $(x,y,z)$ of the equation
$\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$
2017 Romania National Olympiad, 1
Consider the set
$$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$
a) Show that the set $M$ contains no integer.
b) Find the smallest and the largest element of $M$
2010 Lithuania National Olympiad, 4
Decimal digits $a,b,c$ satisfy
\[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \]
where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.
2015 Purple Comet Problems, 12
The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.
Durer Math Competition CD 1st Round - geometry, 2010.D3
Prove that the diagonals of a quadrilateral are perpendicular to each other if and only if the midpoints of it's sides lie on a circle.
2019-IMOC, G3
Given a scalene triangle $\vartriangle ABC$ has orthocenter $H$ and circumcircle $\Omega$. The tangent lines passing through $A,B,C$ are $\ell_a,\ell_b,\ell_c$. Suppose that the intersection of $\ell_b$ and $\ell_c$ is $D$. The foots of $H$ on $\ell_a,AD$ are $P,Q$ respectively. Prove that $PQ$ bisects segment $BC$.
[img]https://4.bp.blogspot.com/-iiQoxMG8bEs/XnYNK7R8S3I/AAAAAAAALeY/FYvSuF6vQQsofASnXJUgKZ1T9oNnd-02ACK4BGAYYCw/s400/imoc2019g3.png[/img]
2002 Vietnam National Olympiad, 3
For a positive integer $ n$, consider the equation $ \frac{1}{x\minus{}1}\plus{}\frac{1}{4x\minus{}1}\plus{}\cdots\plus{}\frac{1}{k^2x\minus{}1}\plus{}\cdots\plus{}\frac{1}{n^2x\minus{}1}\equal{}\frac{1}{2}$.
(a) Prove that, for every $ n$, this equation has a unique root greater than $ 1$, which is denoted by $ x_n$.
(b) Prove that the limit of sequence $ (x_n)$ is $ 4$ as $ n$ approaches infinity.
2001 China Team Selection Test, 3
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
2011 Iran MO (3rd Round), 3
We define the polynomial $f(x)$ in $\mathbb R[x]$ as follows:
$f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0$
Prove that there exists an $i$ in the set $\{1,....,n\}$ such that we have
$|f(i)|\ge \frac{n!}{\dbinom{n}{i}}$.
[i]proposed by Mohammadmahdi Yazdi[/i]
2010 All-Russian Olympiad Regional Round, 11.1
Each leg of a right triangle is increased by one. Could its hypotenuse increase by more than $\sqrt2$?
1982 IMO Longlists, 21
Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality
\[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]
2021 Kyiv City MO Round 1, 10.4
Positive real numbers $a, b, c$ satisfy $a^2 + b^2 + c^2 + a + b + c = 6$. Prove the following inequality:
$$2(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}) \geq 3 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$
[i]Proposed by Oleksii Masalitin[/i]
1998 AIME Problems, 5
Given that $A_k=\frac{k(k-1)}2\cos\frac{k(k-1)\pi}2,$ find $|A_{19}+A_{20}+\cdots+A_{98}|.$
V Soros Olympiad 1998 - 99 (Russia), 11.5
Find the smallest value of the expression
$$(x -y)^2 + (z - u)^2,$$
if $$(x -1)^2 + (y -4)^2 + (z-3)^2 + (u-2)^2 = 1.$$
KoMaL A Problems 2017/2018, A. 722
The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.
2008 Princeton University Math Competition, A5/B8
Let $H_k =\Sigma_{i=1}^k \frac{1}{i}$ for all positive integers $k$. Find an closed-form expression for $\Sigma_{i=1}^k H_i$ in terms of $n$ and $H_n$.
1991 Arnold's Trivium, 59
Investigate the existence and uniqueness of the solution of the problem $yu_x = xu_y, u|_{x=1} =\cos y$ in a neighbourhood of the point $(1, y_0)$.
2002 AMC 10, 2
For the nonzero numbers $ a$, $ b$, $ c$, define
\[(a,b,c)\equal{}\frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}.\]
Find $ (2,12,9)$.
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
2004 Denmark MO - Mohr Contest, 5
Determine for which natural numbers $n$ you can cover a $2n \times 2n$ chessboard with non-overlapping $L$ pieces. An $L$ piece covers four spaces and has appearance like the letter $L$. The piece may be rotated and mirrored at will.
2006 All-Russian Olympiad Regional Round, 10.5
Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$
1992 Poland - First Round, 4
Determine all functions $f: R \longrightarrow R$ such that
$f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$
2005 Today's Calculation Of Integral, 41
Evaluate
\[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]