Found problems: 85335
1991 Greece National Olympiad, 3
In how many ways can we construct a square with dimensions $4\times 4$ using $4$ white, $4$ green , $4$ red and 4 $blue$ squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .
2006 Canada National Olympiad, 3
In a rectangular array of nonnegative reals with $m$ rows and $n$ columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that $m=n$.
2023 Bulgarian Autumn Math Competition, 11.2
The points $A_1, B_1, C_1$ are chosen on the sides $BC, CA, AB$ of a triangle $ABC$ so that $BA_1=BC_1$ and $CA_1=CB_1$. The lines $C_1A_1$ and $A_1B_1$ meet the line through $A$, parallel to $BC$, at $P, Q$. Let the circumcircles of the triangles $APC_1$ and $AQB_1$ meet at $R$. Given that $R$ lies on $AA_1$, show that $R$ lies on the incircle of $ABC$.
1994 Austrian-Polish Competition, 7
Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.
2021 CCA Math Bonanza, TB3
In a party of $2020$ people, some pairs of people are friends. We say that a given person's [i]popularity[/i] is the size of the largest group of people in the party containing them with the property that every pair of people in that group is friends. A person has popularity number $1$ if they have no friends. What is the largest possible number of distinct popularities in the party?
[i]2021 CCA Math Bonanza Tiebreaker Round #3[/i]
2017 Sharygin Geometry Olympiad, 8
10.8 Suppose $S$ is a set of points in the plane, $|S|$ is even; no three points of $S$ are collinear. Prove that $S$ can be partitioned into two sets $S_1$ and $S_2$ so that their convex hulls have equal number of vertices.
1984 Miklós Schweitzer, 4
[b]4.[/b] Let $x_1 , x_2 , y_1 , y_2 , z_1 , z_2 $ be transcendental numbers. Suppose that any 3 of them are algebraically independent, and among the 15 four-tuples on $\{x_1 , x_2 , y_1, y_2 \}$, $\{ x_1 , x_2 , z_1 , z_2 \} $ and $ \{y_1 , y_2 , z_1 , z_2 \} $ are algebraically dependent. Prove that there exists a transcendental number $t$ that depends algebraically on each of the pairs $\{ x_1 , x_2\}$ , $\{ y_1 , y_2 \}$, and $\{ z_1 , z_2 \}$. ([b]A.37[/b])
[L. Lovász]
1999 Putnam, 4
Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.
2004 China Team Selection Test, 2
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.
Prove that $ M$, $ N$, $ O$ are collinear.
2012 ELMO Shortlist, 6
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$.
[i]Calvin Deng.[/i]
2011 Hanoi Open Mathematics Competitions, 11
Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$. Calculate the area of the quadrilateral.
2012 IMC, 5
Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients.
[i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]
2009 Sharygin Geometry Olympiad, 1
Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}} \ge 6r$
(D.Shvetsov)
2005 Spain Mathematical Olympiad, 2
Let $r,s,u,v$ be real numbers. Prove that:
$$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$
2004 China Team Selection Test, 2
There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.
2014 Baltic Way, 17
Do there exist pairwise distinct rational numbers $x, y$ and $z$ such that \[\frac{1}{(x - y)^2}+\frac{1}{(y - z)^2}+\frac{1}{(z - x)^2}= 2014?\]
1979 Czech And Slovak Olympiad IIIA, 3
If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it.
[hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]
2020 LMT Fall, A3
Find the value of $\left\lfloor \frac{1}{6}\right\rfloor+\left\lfloor\frac{4}{6}\right\rfloor+\left\lfloor\frac{9}{6}\right\rfloor+\dots+\left\lfloor\frac{1296}{6}\right\rfloor$.
[i]Proposed by Zachary Perry[/i]
LMT Team Rounds 2010-20, B10
In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?
2006 Bulgaria Team Selection Test, 3
[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$
[i] Ivan Landgev[/i]
2019 Peru Cono Sur TST, P2
Let $AB$ be a diameter of a circle $\Gamma$ with center $O$. Let $CD$ be a chord where $CD$ is perpendicular to $AB$, and $E$ is the midpoint of $CO$. The line $AE$ cuts $\Gamma$ in the point $F$, the segment $BC$ cuts $AF$ and $DF$ in $M$ and $N$, respectively. The circumcircle of $DMN$ intersects $\Gamma$ in the point $K$. Prove that $KM=MB$.
2018 IFYM, Sozopol, 6
There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point.
a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least
$\lfloor \frac{(a-1)(a-2)}{10} \rfloor$
intersection points.
b) Find all $a$ for which the evaluation in a) is the best possible.
Kyiv City MO Seniors 2003+ geometry, 2017.11.5.1
The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.
2001 AMC 10, 25
How many positive integers not exceeding $ 2001$ are multiples of $ 3$ or $ 4$ but not $ 5$?
$ \textbf{(A)}\ 768 \qquad
\textbf{(B)}\ 801 \qquad
\textbf{(C)}\ 934 \qquad
\textbf{(D)}\ 1067 \qquad
\textbf{(E)}\ 1167$
1990 IMO Shortlist, 26
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.