Found problems: 85335
2018 Turkey MO (2nd Round), 5
Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which
$$
({\rm gcd}(a_i,a_j))^2|a_k.
$$
2003 Baltic Way, 1
Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil
\[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]
2023 Macedonian Team Selection Test, Problem 2
Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that
$$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$
[i]Authored by Nikola Velov[/i]
1976 Miklós Schweitzer, 1
Assume that $ R$, a recursive, binary relation on $ \mathbb{N}$ (the set of natural numbers), orders $ \mathbb{N}$ into type $ \omega$. Show that if $ f(n)$ is the $ n$th element of this order, then $ f$ is not necessarily recursive.
[i]L. Posa[/i]
2013 NIMO Problems, 1
Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
[i]Proposed by Evan Chen[/i]
2014 PUMaC Geometry A, 6
$\triangle ABC$ has side lengths $AB=15$, $BC=34$, and $CA=35$. Let the circumcenter of $ABC$ be $O$. Let $D$ be the foot of the perpendicular from $C$ to $AB$. Let $R$ be the foot of the perpendicular from $D$ to $AC$, and let $W$ be the perpendicular foot from $D$ to $BC$. Find the area of quadrilateral $CROW$.
2005 iTest, 9
Find the area of the triangle with vertices of $(1,2)$, $(1,10)$, and $(5, 5)$.
2021 Science ON all problems, 1
Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$.
$\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$.
$\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that
$$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$
Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$.
[i] (Stefan Obadă)[/i]
2011 Tournament of Towns, 2
Several guests at a round table are eating from a basket containing $2011$ berries. Going in clockwise direction, each guest has eaten either twice as many berries as or six fewer berries than the next guest. Prove that not all the berries have been eaten.
1996 May Olympiad, 2
Joining $15^3 = 3375$ cubes of $1$ cm$^3$, bodies with a volume of $3375$ cm$^3$ can be built. Indicate how two bodies $A$ and $B$ are constructed with $3375$ cubes each and such that the lateral surface of $B$ is $10$ times the lateral surface of $A$.
2018 Stanford Mathematics Tournament, 6
In $\vartriangle AB$C, $AB = 3$, $AC = 6,$ and $D$ is drawn on $BC$ such that $AD$ is the angle bisector of $\angle BAC$. $D$ is reflected across $AB$ to a point $E$, and suppose that $AC$ and $BE$ are parallel. Compute $CE$.
2019 District Olympiad, 2
Let $n \in \mathbb{N},n \ge 2,$ and $A,B \in \mathcal{M}_n(\mathbb{R}).$ Prove that there exists a complex number $z,$ such that $|z|=1$ and $$\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B),$$ where $\Re(w)$ is the real part of the complex number $w.$
2016 Regional Olympiad of Mexico Southeast, 1
In a circumference there are $99$ natural numbers. If $a$ and $b$ are two consecutive numbers in the circle, then they must satisfies one of the following conditions: $a-b=1, a-b=2$ or $\frac{a}{b}=2$. Prove that, in the circle exists a number multiple of $3$.
2017 China Team Selection Test, 4
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
1998 Iran MO (2nd round), 2
Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.
2009 AMC 10, 25
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$
2003 China Girls Math Olympiad, 7
Let the sides of a scalene triangle $ \triangle ABC$ be $ AB \equal{} c,$ $ BC \equal{} a,$ $ CA \equal{}b,$ and $ D, E , F$ be points on $ BC, CA, AB$ such that $ AD, BE, CF$ are angle bisectors of the triangle, respectively. Assume that $ DE \equal{} DF.$ Prove that
(1) $ \frac{a}{b\plus{}c} \equal{} \frac{b}{c\plus{}a} \plus{} \frac{c}{a\plus{}b}$
(2) $ \angle BAC > 90^{\circ}.$
2012 India National Olympiad, 5
Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.
2014 Contests, 2
Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$?
[i]Proposed by Evan Chen[/i]
1982 National High School Mathematics League, 11
Length of edges of regular triangle $ABC$ are $4$, $D\in BC,E\in CA,F\in AB$, satisfying: $|AE|=|BF|=|CD|=1$. $BE\cap CF=R, CF\cap AD=Q, AD\cap BE=S$. $P$ is a point inside $\triangle RQS$ or on its sides. Note that $x=d(P,BC),y=d(P,CA),z=d(P,AB)$.
[b](a)[/b] $xyz$ get its minumum value when $P=R$ (or$Q,S$).
[b](b)[/b] Calculate the minumum value of $xyz$.
1987 China Team Selection Test, 3
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
2004 Uzbekistan National Olympiad, 1
Solve the equation:
$[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$
2020 EGMO, 3
Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent.
Prove that the (interior) angle bisectors of $\angle B, ~\angle D, $ and $\angle F$ must also be concurrent.
[i]Note that $\angle A = \angle FAB$. The other interior angles of the hexagon are similarly described.[/i]
2004 Austria Beginners' Competition, 2
For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?
2005 Today's Calculation Of Integral, 25
Let $|a|<\frac{\pi}{2}$.
Evaluate
\[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]