Found problems: 85335
2024 Iranian Geometry Olympiad, 1
In the figure below points $A,B$ are the centers of the circles $\omega_1, \omega_2$. Starting from the line $BC$ points $E,F,G,H,I$ are obtained respectively. Find the angle $\angle IBE$.
2003 District Olympiad, 3
A grid consists of $2n$ vertical and $2n$ horizontal lines, each group disposed at equal distances. The lines are all painted in red and black, such that exactly $n$ vertical and $n$ horizontal lines are red.
Find the smallest $n$ such that for any painting satisfying the above condition, there is a square formed by the intersection of two vertical and two horizontal lines, all of the same colour.
2011 Canadian Students Math Olympiad, 3
Find the largest $C \in \mathbb{R}$ such that
\[\frac{x+z}{(x-z)^2} +\frac{x+w}{(x-w)^2} +\frac{y+z}{(y-z)^2}+\frac{y+w}{(y-w)^2} + \sum_{cyc} \frac{1}{x} \ge \frac{C}{x+y+z+w}\]
where $x,y,z,w \in \mathbb{R^+}$.
[i]Author: Hunter Spink[/i]
2012 Indonesia TST, 3
Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.
1995 Tournament Of Towns, (470) 4
A journalist is looking for a person $Z$ at a meeting of $n$ persons. He has been told that $Z$ knows all the other people at the meeting but none of them knows $Z$. The journalist may ask any person about any other person: “Do you know that person?” One person can be questioned many times. All answers are truthful.
(a) Can the journalist always find $Z$ by asking less than $n$ questions?
(b) What is the minimal number of questions which are needed to find $Z$?
(G Galperin)
1974 Putnam, A5
Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.
Ukraine Correspondence MO - geometry, 2005.7
Let $O$ be the point of intersection of the diagonals of the trapezoid $ABCD$ with the bases $AB$ and $CD$. It is known that $\angle AOB = \angle DAB = 90^o$. On the sides $AD$ and $BC$ take the points $E$ and $F$ so that $EF\parallel AB$ and $EF = AD$. Find the angle $\angle AOE$.
2020 Caucasus Mathematical Olympiad, 7
In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.
2007 Irish Math Olympiad, 1
Find all prime numbers $ p$ and $ q$ such that $ p$ divides $ q\plus{}6$ and $ q$ divides $ p\plus{}6$.
2017 Putnam, B1
Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every real number $\lambda\ne 0$ and every point $P$ not on $L_1$ or $L_2,$ there exist points $A_1$ on $L_1$ and $A_2$ on $L_2$ such that $\overrightarrow{PA_2}=\lambda\overrightarrow{PA_1}.$
1986 Miklós Schweitzer, 5
Prove that existence of a constant $c$ with the following property: for every composite integer $n$, there exists a group whose order is divisible by $n$ and is less than $n^c$, and that contains no element of order $n$. [P. P. Palfy]
2014 District Olympiad, 1
Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying
\[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]
2001 National Olympiad First Round, 26
Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. If Berk can guarantee to guess Ayca's number in $n$ guesses, what is the smallest possible value of $n$?
$
\textbf{(A)}\ 9
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 15
\qquad\textbf{(E)}\ 20
$
2011 Turkey Team Selection Test, 3
Let $p$ be a prime, $n$ be a positive integer, and let $\mathbb{Z}_{p^n}$ denote the set of congruence classes modulo $p^n.$ Determine the number of functions $f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}$ satisfying the condition
\[ f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} \]
for all $a,b \in \mathbb{Z}_{p^n}.$
2021 MIG, 8
A square's area is equal to the perimeter of a $15$ by $17$ rectangle. What is this square's perimeter?
$\textbf{(A) }20\qquad\textbf{(B) }32\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }56$
2015 Peru IMO TST, 14
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that:
\[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}.
\]
Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
1987 IMO Longlists, 26
Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$, then
\[x + y + z \leq xyz + 2.\]
1984 Bulgaria National Olympiad, Problem 4
Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation
$$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$are positive and real. Prove that these roots are all equal.
VI Soros Olympiad 1999 - 2000 (Russia), 9.1
In the television program “Field of Miracles,” the presenter played the prize as follows. The player was shown three boxes, one of which contained a prize. The player pointed to one of the boxes, after which the leader opened one of the other two remaining boxes, which turned out to be empty. After this, the player could either insist on the original choice, or change it and choose the third box. In what case does his chance of winning increase? (There are three possible answers: both boxes are equal, it is better to keep the original choice, it is better to change it. Try to justify your answer.)
1989 Federal Competition For Advanced Students, 3
Let $ a$ be a real number. Prove that if the equation $ x^2\minus{}ax\plus{}a\equal{}0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).$
2008 International Zhautykov Olympiad, 3
Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that
$ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.
2019 Serbia Team Selection Test, P4
A trader owns horses of $3$ races, and exacly $b_j$ of each race (for $j=1,2,3$). He want to leave these horses heritage to his $3$ sons. He knowns that the boy $i$ for horse $j$ (for $i,j=1,2,3$) would pay $a_{ij}$ golds, such that for distinct $i,j$ holds holds $a_{ii}> a_{ij}$ and $a_{jj} >a_{ij}$.
Prove that there exists a natural number $n$ such that whenever it holds $\min\{b_1,b_2,b_3\}>n$, trader can give the horses to their sons such that after getting the horses each son values his horses more than the other brother is getting, individually.
2023 Chile National Olympiad, 3
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$.
[img]https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png[/img]
PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3[/url]
2020 JBMO TST of France, 4
Let $a_0, a_1,...$ be a sequence of non-negative integers and $b_0, b_1,... $ be a sequence of non-negative integers defined by the following rule:
$b_i=gcd(a_i, a_{i+1})$ for every $i=>0$
Is it possible every positive integer to occur exactly once in the sequence $b_0, b_1,... $
2015 Cuba MO, 2
Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,