Found problems: 85335
PEN O Problems, 35
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that
\[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\]
where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.
2012 Turkey MO (2nd round), 4
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
1989 AMC 12/AHSME, 30
Suppose that $7$ boys and $13$ girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $GBBGGGBGBGGGBGBGGBGG$ we have $S=12$. The average value of $S$ (if all possible orders of the 20 people are considered) is closest to
$ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13 $
2021 AMC 10 Fall, 24
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }20$
2018 ASDAN Math Tournament, 3
In $\vartriangle ABC$, $AC > AB$. $B$ is reflected across $\overline{AC}$ to a point $D$, and $C$ is reflected across $\overline{AD}$ to a point $E$. Suppose that $AC = 6\sqrt3 + 6$, $BC = 6$, and $\overline{BC} \parallel \overline{AE}$. Compute $AB$.
Kvant 2024, M2812
On the coordinate plane, at some points with integer coordinates, there is a pebble (a finite number of pebbles). It is allowed to make the following move: select a pair of pebbles, take some vector $\vec{a}$ with integer coordinates and then move one of the selected pebbles to vector $\vec{a}$, and the other to the opposite vector $-\vec{a}$; it is forbidden that there should be more than one pebble at one point. Is it always possible to achieve a situation in which all the pebbles lie on the same straight line in a few moves?
[i] K. Ivanov [/i]
2010 Harvard-MIT Mathematics Tournament, 10
Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Segment $PQ$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$, and $A$ is closer to $PQ$ than $B$. Point $X$ is on $\omega_1$ such that $PX\parallel QB$, and point $Y$ is on $\omega_2$ such that $QY\parallel PB$. Given that $\angle APQ=30^\circ$ and $\angle PQA=15^\circ$, find the ratio $AX/AY$.
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.