Found problems: 85335
2014 Contests, 1
In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.
Croatia MO (HMO) - geometry, 2017.7
The point $M$ is located inside the triangle $ABC$. The ray $AM$ intersects the circumcircle of the triangle $MBC$ once more at point $D$, the ray $BM$ intersects the circumcircle of the triangle $MCA$ once more at point $E$, and the ray $CM$ intersects the circumcircle of the triangle $MAB$ once more at point $F$. Prove that holds
$$\frac{AD}{MD}+\frac{BE}{ME} +\frac{CF}{MF}\ge \frac92 $$
2010 Contests, 1
Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$.
(Peter Cameron)
2024 Princeton University Math Competition, A8
Let $a,b,c$ be pairwise coprime integers such a that $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac{N}{a+b+c}$ for some positive integer $N.$ What is the sum of all possible values of $N.$
1995 Spain Mathematical Olympiad, 2
Several paper-made disks (not necessarily equal) are put on the table so that there is some overlapping, but no disk is entirely inside another. The parts that overlap are cut off and removed. Show that the remaining parts cannot be assembled so as to form different disks.
2016 Sharygin Geometry Olympiad, 8
The diagonals of a cyclic quadrilateral meet at point $M$. A circle $\omega$ touches segments $MA$ and $MD$ at points $P,Q$ respectively and touches the circumcircle of $ABCD$ at point $X$. Prove that $X$ lies on the radical axis of circles $ACQ$ and $BDP$.
[i](Proposed by Ivan Frolov)[/i]
2009 Indonesia TST, 2
For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.
2016 AMC 10, 19
In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s$ and $t$ is $1$. What is $r+s+t$?
$\textbf{(A) } 7
\qquad \textbf{(B) } 9
\qquad \textbf{(C) } 12
\qquad \textbf{(D) } 15
\qquad \textbf{(E) } 20$
1966 All Russian Mathematical Olympiad, 083
$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?
2009 China Team Selection Test, 4
Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.
PEN M Problems, 19
A sequence with first two terms equal $1$ and $24$ respectively is defined by the following rule: each subsequent term is equal to the smallest positive integer which has not yet occurred in the sequence and is not coprime with the previous term. Prove that all positive integers occur in this sequence.
2023 MMATHS, 5
We call $\triangle{ABC}$ with centroid $G$ [i]balanced[/i] on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B.$ $\triangle{XYZ},$ with $XY=2023$ and $\angle{ZXY}=120^\circ,$ is balanced on $XY.$ What is the maximum value of $XZ$?
2000 Stanford Mathematics Tournament, 13
How many permutations of $123456$ have exactly one number in the correct place?
2020 Romanian Master of Mathematics Shortlist, A2
Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$?
[i]Adam Ardeishar and Michael Ren[/i]
2016 European Mathematical Cup, 2
Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$.
Proposed by Steve Dinh
2018 Sharygin Geometry Olympiad, 17
Let each of circles $\alpha, \beta, \gamma$ touches two remaining circles externally, and all of them touche a circle $\Omega$ internally at points $A_1, B_1, C_1$ respectively. The common internal tangent to $\alpha$ and $\beta$ meets the arc $A_1B_1$ not containing $C_1$ at point $C_2$. Points $A_2$, $B_2$ are defined similarly. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ concur.
2003 Korea - Final Round, 3
There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$).
1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$.
2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.
Kvant 2020, M2603
For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers.
Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.
R. Salimov
2017 AMC 12/AHSME, 17
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
1978 Chisinau City MO, 163
On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.
2004 VJIMC, Problem 2
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
2021 CMIMC, 2
You are initially given the number $n=1$. Each turn, you may choose any positive divisor $d\mid n$, and multiply $n$ by $d+1$. For instance, on the first turn, you must select $d=1$, giving $n=1\cdot(1+1)=2$ as your new value of $n$. On the next turn, you can select either $d=1$ or $2$, giving $n=2\cdot(1+1)=4$ or $n=2\cdot(2+1)=6$, respectively, and so on.
Find an algorithm that, in at most $k$ steps, results in $n$ being divisible by the number $2021^{2021^{2021}} - 1$.
An algorithm that completes in at most $k$ steps will be awarded:
1 pt for $k>2021^{2021^{2021}}$
20 pts for $k=2021^{2021^{2021}}$
50 pts for $k=10^{10^4}$
75 pts for $k=10^{10}$
90 pts for $k=10^5$
95 pts for $k=6\cdot10^4$
100 pts for $k=5\cdot10^4$
2014 National Olympiad First Round, 27
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 120
\qquad\textbf{(C)}\ 165
\qquad\textbf{(D)}\ 180
\qquad\textbf{(E)}\ 215
$
1999 Romania National Olympiad, 3
Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$
satisfy $|z_3|=|z_4|=1$
JOM 2014, 3.
There is a complete graph $G$ with $4027$ vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs $(r, b)$ such that Ivan can paint the edges so that every vertex is connected to exactly $r$ red edges and $b$ blue edges.