Found problems: 85335
1997 Brazil Team Selection Test, Problem 1
Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.
2018 Federal Competition For Advanced Students, P2, 5
On a circle $2018$ points are marked. Each of these points is labeled with an integer.
Let each number be larger than the sum of the preceding two numbers in clockwise order.
Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers.
[i](Proposed by Walther Janous)[/i]
2019 CCA Math Bonanza, T7
How many ordered triples $\left(a,b,c\right)$ of positive integers are there such that at least two of $a,b,c$ are prime and $abc=11\left(a+b+c\right)$?
[i]2019 CCA Math Bonanza Team Round #7[/i]
Brazil L2 Finals (OBM) - geometry, 2000.3
A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.
2017 Dutch IMO TST, 4
Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM = \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$.
$(a)$ Show that $CM$ is tangent to $\omega$.
$(b)$ Show that the lines $NX$ and $AC$ intersect on $\omega$
1993 AMC 12/AHSME, 15
For how many values of $n$ will an $n$-sided regular polygon have interior angles with integral degree measures?
$ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $
2022 Harvard-MIT Mathematics Tournament, 4
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$|S| +\ min(S) \cdot \max (S) = 0.$$
2016 Switzerland Team Selection Test, Problem 8
Let $ABC$ be a triangle with $AB \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.
ICMC 2, 2
This question, again, comprises two independent parts.
(i) Show that if \((k+1)\) integers are chosen from \(\left\{1,2,3,...,2k+1\right\}\), then among the chosen integers there are always two that are coprime.
(ii) Let \(A=\left\{1,2,\ldots,n\right\}.\) Prove that if \(n>11\) then there is a bijective map \(f: A\to A\) with the property that, for every \(a\in A\), exactly one of \(f(f(f(f(a))))=a\) and \(f(f(f(f(f(a)))))=a\) holds.
2008 AIME Problems, 12
There are two distinguishable flagpoles, and there are $ 19$ flags, of which $ 10$ are identical blue flags, and $ 9$ are identical green flags. Let $ N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when $ N$ is divided by $ 1000$.
Novosibirsk Oral Geo Oly VIII, 2021.4
Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.
2002 Denmark MO - Mohr Contest, 5
Homer Grog has written the numbers $1, 3, 4, 5, 7, 9, 11, 13, 15,17$, one number on each note. He arranges the bills in a circle and tries to get the largest sum $S$ of the numbers of three consecutive bills to be the least possible. What is the smallest value $S$ can assume?
2011 China Northern MO, 7
In $\triangle ABC$ , then \[\frac{1}{1+\cos^2 A+\cos^2 B}+\frac{1}{1+\cos^2 B+\cos^2 C}+\frac{1}{1+\cos^2 C+\cos^2 A}\le 2\]
1992 IMO Longlists, 81
Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$
1986 Swedish Mathematical Competition, 6
The interval $[0,1]$ is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least $1/2$.
2012 IMAR Test, 2
Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.
2015 Oral Moscow Geometry Olympiad, 3
$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.
2016 CHMMC (Fall), 4
Line segments $m$ and $n$ both have length $2$ and bisect each other at an angle of $60^o$, as shown. A point $X$ is placed at uniform random position along $n$, and a point $Y$ is placed at a uniform random position along $m$. Find the probability that the distance between $X$ and $Y$ is less than $\frac12$.
2016 CHMMC (Fall), 11
Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?
2022 CMIMC, 1.7
In a class of $12$ students, no two people are the same height. Compute the total number of ways for the students to arrange themselves in a line such that:
[list]
[*] for all $1 < i < 12$, the person in the $i$-th position (with the leftmost position being $1$) is taller than exactly $i\pmod 3$ of their adjacent neighbors, and
[*] the students standing at positions which are multiples of $3$ are strictly increasing in height from left to right.
[/list]
[i]Proposed by Nancy Kuang[/i]
2013 AMC 12/AHSME, 5
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $
2016 India Regional Mathematical Olympiad, 6
Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.
2007 Rioplatense Mathematical Olympiad, Level 3, 6
Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets .
Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .
2008 239 Open Mathematical Olympiad, 1
Composite numbers $a$ and $b$ have equal number of divisors. All proper divisors of $a$ were written in ascending order and all proper divisors of $b$ were written under them in ascending order, then the numbers that are below each other were added together. It turned out that the resulting numbers formed a set of all proper divisors of a certain number. What are the smallest values that $a$ and $b$ take?
2018 Bulgaria EGMO TST, 1
In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a loss is worth $0$ points. Determine the smallest positive integer $n$ for which it is possible that after the $n$-th match all teams have a different number of points and each team has a non-zero number of points.