Found problems: 85335
2011 All-Russian Olympiad, 3
A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
2023 LMT Fall, 9
In triangle $ABC$, let $O$ be the circumcenter and let $G$ be the centroid. The line perpendicular to $OG$ at $O $ intersects $BC$ at $M$ such that $M$, $G$, and $A$ are collinear and $OM = 3$. Compute the area of $ABC$, given that $OG = 1$.
2008 Harvard-MIT Mathematics Tournament, 12
Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?
2003 India Regional Mathematical Olympiad, 6
Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$.
2023 AMC 8, 21
Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done?
$\textbf{(A) }0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$
2010 All-Russian Olympiad Regional Round, 10.5
Non-zero numbers $a, b, c$ are such that $ax^2+bx+c > cx$ for any $x$.
Prove that $cx^2-bx + a > cx-b$ for any $x$.
2007 iTest Tournament of Champions, 2
Al and Bill play a game involving a fair six-sided die. The die is rolled until either there is a number less than $5$ rolled on consecutive tosses, or there is a number greater than $4$ on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $m/n$ is the probability that Bill wins. Find the value of $m+n$.
2021 Germany Team Selection Test, 3
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2019 Iran MO (3rd Round), 1
Hossna is playing with a $m*n$ grid of points.In each turn she draws segments between points with the following conditions.
**1.** No two segments intersect.
**2.** Each segment is drawn between two consecutive rows.
**3.** There is at most one segment between any two points.
Find the maximum number of regions Hossna can create.
2006 MOP Homework, 4
Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.
2019 Yasinsky Geometry Olympiad, p2
Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$.
1983 Poland - Second Round, 5
The bisectors of the angles $ CAB, ABC, BCA $ of the triangle $ ABC $ intersect the circle circumcribed around this triangle at points $ K, L, M $, respectively. Prove that $$ AK+BL+CM > AB+BC+CA.$$
1987 Romania Team Selection Test, 7
Determine all positive integers $n$ such that $n$ divides $3^n - 2^n$.
1979 Spain Mathematical Olympiad, 3
Prove the equality
$${n \choose 0}^2+ {n \choose 1}^2+ {n \choose 2}^2+...+{n \choose n}^2={2n \choose n}$$
2018 Costa Rica - Final Round, LRP4
On a $30\times 30$ board both rows $ 1$ to $30$ and columns are numbered, in addition, to each box is assigned the number $ij$, where the box is in row $i$ and column $j$.
$N$ columns and $m$ rows are chosen, where $1 <n$ and $m <30$, and the cells that are simultaneously in any of the rows and in any of the selected columns are painted blue. They paint the others red .
(a) Prove that the sum of the numbers in the blue boxes cannot be prime.
(b) Can the sum of the numbers in the red cells be prime?
2009 Indonesia TST, 1
Let $ ABC$ be a triangle. A circle $ P$ is internally tangent to the circumcircle of triangle $ ABC$ at $ A$ and tangent to $ BC$ at $ D$. Let $ AD$ meets the circumcircle of $ ABC$ agin at $ Q$. Let $ O$ be the circumcenter of triangle $ ABC$. If the line $ AO$ bisects $ \angle DAC$, prove that the circle centered at $ Q$ passing through $ B$, circle $ P$, and the perpendicular line of $ AD$ from $ B$, are all concurrent.
2020 Balkan MO Shortlist, G5
Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle A = 45^o$. Its circumcircle $(c)$ has center $O, M$ is the midpoint of $BC$ and $D$ is the foot of the perpendicular from $C$ to $AB$. With center $C$ and radius $CD$ we draw a circle which internally intersects $AC$ at the point $F$ and the circle $(c)$ at the points $Z$ and $E$, such that $Z$ lies on the small arc $BC$ and $E$ on the small arc $AC$. Prove that the lines $ZE$, $CO$, $FM$ are concurrent.
[i]Brazitikos Silouanos, Greece[/i]
2009 Greece National Olympiad, 4
Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$
If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold \[z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0\] prove that
[b]a)[/b] Triangle $A_1A_3A_5$ is equilateral
[b]b)[/b] \[|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.\]
1995 Baltic Way, 16
In the triangle $ABC$, let $\ell$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of $AB$ parallel to $\ell$ meets $AC$ at $E$. Determine $|CE|$, if $|AC|=7$ and $|CB|=4$.
1999 Argentina National Olympiad, 3
In a trick tournament $2k$ people sign up. All possible matches are played with the condition that in each match, each of the four players knows his partner and does not know any of his two opponents. Determine the maximum number of matches that can be in such a tournament.
2023 CMIMC Combo/CS, 8
How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x))+f(x)+x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}?$
[i]Proposed by Kyle Lee[/i]
2006 District Olympiad, 4
We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point.
b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.
2018 Canadian Mathematical Olympiad Qualification, 5
A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$. Prove that infinitely many multiples of $n$ are palindromes.
1982 IMO Shortlist, 7
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
1984 AIME Problems, 4
Let $S$ be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in $S$ is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in $S$?