Found problems: 85335
2023 Tuymaada Olympiad, 6
In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality
\[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]
2004 Abels Math Contest (Norwegian MO), 4
Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends.
(a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones.
(b) Prove that this is not necessarily true with less than $n^2/4$ types.
2012 India PRMO, 17
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
2006 AMC 10, 7
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
[asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$
2004 Nicolae Coculescu, 4
Let be a matrix $ A\in\mathcal{M}_2(\mathbb{R}) $ having the property that the numbers $ \det (A+X) ,\det (A^2+X^2) ,\det (A^3+X^3) $ are (in this order) in geometric progression, for any matrix $ X\in\mathcal{M}_2(\mathbb{R}) . $
Prove that $ A=0. $
[i]Marius Ghergu[/i]
1961 Putnam, B4
Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms:
$$\sum_{i<j}|x_i -x_j |.$$
1984 Canada National Olympiad, 3
An integer is digitally divisible if both of the following conditions are fulfilled:
$(a)$ None of its digits is zero;
$(b)$ It is divisible by the sum of its digits
e.g. $322$ is digitally divisible. Show that there are infinitely many digitally divisible integers.
2023 Oral Moscow Geometry Olympiad, 4
Let $I$ be the incenter of triangle $ABC$, tangent to sides $AB$ and $AC$ at points $E$ and $F$, respectively. The lines through $E$ and $F$ parallel to $AI$ intersect lines $BI$ and $CI$ at points $P$ and $Q$, respectively. Prove that the center of the circumcircle of triangle $IPQ$ lies on line $BC$.
2012 AIME Problems, 7
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the $1000^{th}$ number in $S$. Find the remainder when $N$ is divided by $1000$.
2007 Kazakhstan National Olympiad, 1
Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.
2022 LMT Spring, 6
For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.
2008 Indonesia TST, 2
Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$.
Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$
for all positive integers $n$.
2002 Austrian-Polish Competition, 2
Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.
2010 IFYM, Sozopol, 8
Find all polynomials $f(x)$ with integer coefficients and leading coefficient equal to 1, for which $f(0)=2010$ and for each irrational $x$, $f(x)$ is also irrational.
2004 German National Olympiad, 1
Find all real numbers $x,y$ satisfying the following system of equations
\begin{align*}
x^4 +y^4 & =17(x+y)^2 \\
xy & =2(x+y).
\end{align*}
2003 AMC 8, 4
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
1997 Hungary-Israel Binational, 3
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
2020 CHMMC Winter (2020-21), 1
Triangle $ABC$ has circumcircle $\Omega$. Chord $XY$ of $\Omega$ intersects segment $AC$ at point $E$ and segment $AB$ at point $F$ such that $E$ lies between $X$ and $F$. Suppose that $A$ bisects arc $\widehat{XY}$. Given that $EC = 7, FB = 10, AF = 8$, and $YF - XE = 2$, find the perimeter of triangle $ABC$.
2003 Regional Competition For Advanced Students, 1
Find the minimum value of the expression $ \frac{a\plus{}1}{a(a\plus{}2)}\plus{}\frac{b\plus{}1}{b(b\plus{}2)}\plus{}\frac{c\plus{}1}{c(c\plus{}2)}$, where $ a,b,c$ are positive real numbers with $ a\plus{}b\plus{}c \le 3$.
1974 Dutch Mathematical Olympiad, 1
A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.
2004 Baltic Way, 6
A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?
2002 China Team Selection Test, 2
Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds
\[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \]
where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.
2019 Polish Junior MO Finals, 1.
Let $a$, $b$ be the positive integers greater than $1$. Prove that if
$$
\frac{a}{b},\; \frac{a-1}{b-1}
$$
differ by 1, then both are integers.
2009 Balkan MO Shortlist, G1
In the triangle $ABC, \angle BAC$ is acute, the angle bisector of $\angle BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$, and $\angle ADB = 45^o$. Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$. Point $Q$ lies on the ray $DP$ such that $DQ = DK$. The perpendicular at $P$ to $AC$ meets $KD$ at $L$. Prove that $PL^2 = DQ \cdot PQ$.
1996 APMO, 2
Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that
\[ 2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \]