Found problems: 85335
1993 Miklós Schweitzer, 10
Let $U_1 , U_2 , U_3$ be iid random variables on [0,1], which in order of magnitude, $U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}$. Let $\alpha, p_1 , p_2 , p_3 \in [0,1]$ such that $P(U_j ^ {\ast} \ge p_j)= \alpha$ ( j = 1,2,3). Prove that
$$P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha$$
LMT Team Rounds 2021+, 6
Call a polynomial $p(x)$ with positive integer roots [i]corrupt[/i] if there exists an integer that cannot be expressed as a sum of (not necessarily positive) multiples of its roots. The polynomial $A(x)$ is monic, corrupt, and has distinct roots. As well, $A(0)$ has $7$ positive divisors. Find the least possible value of $|A(1)|$.
1998 Turkey MO (2nd round), 3
Some of the vertices of unit squares of an $n\times n$ chessboard are colored so that any $k\times k$ ( $1\le k\le n$) square consisting of these unit squares has a colored point on at least one of its sides. Let $l(n)$ denote the minimum number of colored points required to satisfy this condition. Prove that $\underset{n\to \infty }{\mathop \lim }\,\frac{l(n)}{{{n}^{2}}}=\frac{2}{7}$.
2018 India IMO Training Camp, 2
Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$
2020 AMC 10, 1
What value of $x$ satisfies
$$x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?$$
$\textbf{(A)}\ -\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$
2009 Cuba MO, 6
Let $\omega_1$ and $\omega_2$ be circles that intersect at points $A$ and $B$ and let $O_1$ and $O_2$ be their respective centers. We take $M$ in $\omega_1$ and $N$ in $\omega_2$ on the same side as $B$ with respect to segment $O_1O_2$, such that $MO_1\parallel BO_2$ and $BO_1 \parallel NO_2$. Draw the tangents to $\omega_1$ and $\omega_2$ through $M$ and $N$ respectively, which intersect at $K$. Show that $A$, $B$ and $K$ are collinear.
2011 ELMO Shortlist, 7
Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges.
[i]David Yang.[/i]
2013 Thailand Mathematical Olympiad, 9
Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .
2020 USAMTS Problems, 3:
Find, with proof, all positive integers $n$ with the following property: There are only finitely many positive multiples of $n$ which have exactly $n$ positive divisors
1994 Tournament Of Towns, (434) 4
A rectangular $1$ by $10$ strip is divided into $10$ $1$ by $1$ squares. The numbers $1$, $2$, $3$,$...$, $10$ are placed in the squares in the following way. First the number $1$ is placed in an arbitrary square, then $2$ is placed in a neighbouring square, then $3$ is placed into a free square neighbouring one of the squares occupied earlier, and so on (up to $10$). How many different permutations of $1$,$2$, $3$,$...$, $10$ can one get in this way?
(A Shen)
2019 Oral Moscow Geometry Olympiad, 2
On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.
2014 Contests, Problem 3
Given $n\geq2$, let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$, it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$.
Prove that $|\mathcal{A}| \leq 2^{n-2}.$
2000 May Olympiad, 4
There are pieces in the shape of an equilateral triangle with sides $1, 2, 3, 4, 5$ and $6$ ($50$ pieces of each size). You want to build an equilateral triangle of side $7$ using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?
1973 AMC 12/AHSME, 16
If the sum of all the angles except one of a convex polygon is $ 2190^{\circ}$, then the number of sides of the polygon must be
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 19 \qquad
\textbf{(E)}\ 21$
2019 Danube Mathematical Competition, 2
Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$
for any real numbers $ x,y. $
2020 Ukrainian Geometry Olympiad - April, 2
Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.
Ukrainian From Tasks to Tasks - geometry, 2014.9
On a circle with diameter $AB$ we marked an arbitrary point $C$, which does not coincide with $A$ and $B$. The tangent to the circle at point $A$ intersects the line $BC$ at point $D$. Prove that the tangent to the circle at point $C$ bisects the segment $AD$.
2006 Germany Team Selection Test, 2
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2012 IMO Shortlist, N8
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.
2014 All-Russian Olympiad, 4
In a country of $n$ cities, an express train runs both ways between any two cities. For any train, ticket prices either direction are equal, but for any different routes these prices are different. Prove that the traveler can select the starting city, leave it and go on, successively, $n-1$ trains, such that each fare is smaller than that of the previous fare. (A traveler can enter the same city several times.)
1991 Czech And Slovak Olympiad IIIA, 1
Prove that for any real numbers $p,q,r,\phi$,:
$$\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})$$
2015 Iran Team Selection Test, 3
$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)
2004 Putnam, B1
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers
$c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$
are all integers.
2010 China Western Mathematical Olympiad, 4
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:
(1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$;
(2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$;
(3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$.
Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.
2022 China Team Selection Test, 6
Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$,
\[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \]
Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.