Found problems: 85335
2024 Kurschak Competition, 1
The quadrilateral $ABCD$ is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral $ABCD$. Prove that $ABCD$ is also a cyclic quadrilateral.
IV Soros Olympiad 1997 - 98 (Russia), 9.6
A chord is drawn through the intersection point of the diagonals of an inscribed quadrilateral. It is known that the parts of this chord located outside the quadrilateral have lengths equal to $\frac13$ and $\frac14$ of this chord. In what ratio is this chord divided by the intersection point of the diagonals of the quadrilateral?
2010 Silk Road, 2
Let $N = 2010!+1$. Prove that
a) $N$ is not divisible by $4021$;
b) $N$ is not divisible by $2027,2029,2039$;
c)$ N$ has a prime divisor greater than $2050$.
2014 Iran MO (2nd Round), 2
Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]
1999 AMC 12/AHSME, 22
The graphs of $ y \equal{} \minus{}|x \minus{} a| \plus{} b$ and $ y \equal{} |x \minus{} c| \plus{} d$ intersect at points $ (2,5)$ and $ (8,3)$. Find $ a \plus{} c$.
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 13\qquad
\textbf{(E)}\ 18$
2009 AMC 8, 16
How many $ 3$-digit positive integers have digits whose product equals $ 24$?
$ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$
1967 IMO Shortlist, 1
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
2018 IFYM, Sozopol, 7
On the sides $AC$ and $AB$ of an acute $\triangle ABC$ are chosen points $M$ and $N$ respectively. Point $P$ is an intersection point of the segments $BM$ and $CN$ and point $Q$ is an inner point for the quadrilateral $ANPM$, for which $\angle BQC = 90^\circ$ and $\angle BQP = \angle BMQ$. If the quadrilateral $ANPM$ is inscribed in a circle, prove that $\angle QNC = \angle PQC$.
2021 Israel TST, 1
A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which
\[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\]
where $s(n)$ denotes the sum of digits of $n$ in decimal representation.
Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.
1959 IMO, 5
An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$.
a) Prove that $N$ and $N'$ coincide;
b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$;
c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.
2021 Science ON all problems, 2
Find all pairs $(p,q)$ of prime numbers such that
$$p^q-4~|~q^p-1.$$
[i](Vlad Robu)[/i]
2019 Israel Olympic Revenge, P1
A polynomial $P$ in $n$ variables and real coefficients is called [i]magical[/i] if $P(\mathbb{N}^n)\subset \mathbb{N}$, and moreover the map $P: \mathbb{N}^n \to \mathbb{N}$ is a bijection. Prove that for all positive integers $n$, there are at least
\[n!\cdot (C(n)-C(n-1))\]
magical polynomials, where $C(n)$ is the $n$-th Catalan number.
Here $\mathbb{N}=\{0,1,2,\dots\}$.
2002 VJIMC, Problem 3
Positive numbers $x_1,\ldots,x_n$ satisfy
$$\frac1{1+x_1}+\frac1{1+x_2}+\ldots+\frac1{1+x_n}=1.$$Prove that
$$\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_n}\ge(n-1)\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}+\ldots+\frac1{\sqrt{x_n}}\right).$$
2011 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$
2013 Online Math Open Problems, 8
Suppose that $x_1 < x_2 < \dots < x_n$ is a sequence of positive integers such that $x_k$ divides $x_{k+2}$ for each $k = 1, 2, \dots, n-2$. Given that $x_n = 1000$, what is the largest possible value of $n$?
[i]Proposed by Evan Chen[/i]
2021 Peru Iberoamerican Team Selection Test, P1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
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 $1000$.