Found problems: 85335
Ukraine Correspondence MO - geometry, 2004.8
The extensions of the sides $AB$ and $CD$ of the trapezoid $ABCD$ intersect at point $E$. Denote by $H$ and $G$ the midpoints of $BD$ and $AC$. Find the ratio of the area $AEGH$ to the area $ABCD$.
2000 Poland - Second Round, 6
Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.
1985 IMO, 1
A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.
2009 Estonia Team Selection Test, 5
A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip.
Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and
a) $n = 2008$,
b) $n = 2009$.
1995 Israel Mathematical Olympiad, 6
A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called [i]good [/i] if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$-coloring of the given board is [i]good[/i].
2021 AMC 12/AHSME Spring, 15
A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of groups that can be selected. What is the remainder when $N$ is divided by $100$?
$\textbf{(A)}\ 47 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 95 \qquad\textbf{(E)}\ 96$
2014 China Northern MO, 4
In an election, there are a total of $12$ candidates. An election committee has $6$ members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.
2008 Grigore Moisil Intercounty, 2
Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that
$$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $
[i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]
Kvant 2022, M2718
$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.
2004 India IMO Training Camp, 1
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]
2018 Junior Regional Olympiad - FBH, 5
In triangle $ABC$ length of altitude $CH$, with $H \in AB$, is equal to half of side $AB$. If $\angle BAC = 45^{\circ}$ find $\angle ABC$
2008 Indonesia TST, 2
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.
2012 Kyrgyzstan National Olympiad, 2
Given positive real numbers $ {a_1},{a_2},...,{a_n} $ with $ {a_1}+{a_2}+...+{a_n}= 1 $. Prove that
$ \left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n} $.
2024 CMIMC Algebra and Number Theory, 1
Connor is thinking of a two-digit number $n$, which satisfies the following properties:
[list]
[*] If $n>70$, then $n$ is a perfect square.
[*] If $n>40$, then $n$ is prime.
[*] If $n<80$, then the sum of the digits of $n$ is $14$.
[/list]
What is Connor's number?
[i]Proposed by Connor Gordon[/i]
2005 Taiwan National Olympiad, 1
$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.
2018 JBMO Shortlist, C1
A set $S$ is called [i]neighbouring [/i] if it has the following two properties:
a) $S$ has exactly four elements
b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$.
Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.
2024 Kosovo EGMO Team Selection Test, P3
Let $\triangle ABC$ be a right triangle at the vertex $A$ such that the side $AB$ is shorter than the side $AC$.
Let $D$ be the foot of the altitude from $A$ to $BC$ and $M$ the midpoint of $BC$. Let $E$ be a point on the ray $AB$, outside of the segment $AB$. Line $ED$ intersects the segment $AM$ at the point $F$. Point $H$ is on the side $AC$ such that $\angle EFH=90^{\circ}$. Suppose that $ED=FH$. Find the measure of the angle $\angle AED$.
2023 Belarusian National Olympiad, 11.7
Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $AB$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$
Find all possible values of $\frac{AB+AC}{BC}$
2012 Hanoi Open Mathematics Competitions, 3
[b]Q3.[/b] For any possitive integer $a$, let $\left[ a\right]$ denote the smallest prime factor of $a.$ Which of the following numbers is equal to $\left[ 35 \right]$ ?
\[(A) \; \left[10 \right]; \qquad (B) \; \left[ 15 \right]; \qquad (C ) \; \left[45 \right]; \qquad (D) \; \left[ 55 \right]; \qquad (E) \; \left[75 \right].\]
2000 Harvard-MIT Mathematics Tournament, 14
Define a sequence $<x_n>$ of real numbers by specifying an initial $x_0$ and by the recurrence $x_{n+1}=\frac{1+x_n}{1-x_n}$. Find $x_n$ as a function of $x_0$ and $n$, in closed form. There may be multiple cases.
2003 AIME Problems, 11
An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.
LMT Guts Rounds, 17
Determine the sum of the two largest prime factors of the integer $89! + 90!.$
2018 MIG, 2
The MIG is planning a lottery to give out prizes after the written tests, and the plan is very special. Contestants will be divided into prize groups in order to potentially receive a prize. However, based on the number of contestants, the ideal number of groups don't work. For example, when dividing into $4$ groups, there are $3$ left over. When dividing into $5$ groups, there's $2$ left over. When dividing into $6$ groups, theres $1$ left over. Finally, when dividing into $7$ groups, there are $2$ left over. With the knowledge that there are less than $300$ participants in the MIG, how many participants are there?
2000 Mongolian Mathematical Olympiad, Problem 5
Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities:
\begin{align*}
&1\le v+x-y\le n,\\
&1\le x+y-u\le n,\\
&1\le u+v-y\le n,\\
&1\le v+x-u\le n.
\end{align*}
1988 IMO Longlists, 30
In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.