Found problems: 85335
2023 Junior Balkan Mathematical Olympiad, 4
Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic.
[i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]
2013 Harvard-MIT Mathematics Tournament, 27
Let $W$ be the hypercube $\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}$. The intersection of $W$ and a hyperplane parallel to $x_1+x_2+x_3+x_4=0$ is a non-degenerate $3$-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
I Soros Olympiad 1994-95 (Rus + Ukr), 9.6
A circle can be drawn around the quadrilateral $ABCD$. $K$ is a point on the diagonal $BD$ . The straight line $CK$ intersects the side $AD$ at the point $M$. Prove that the circles circumscribed around the triangles $BCK$ and $ACM$ are tangent.
LMT Speed Rounds, 2011.17
Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?
2016 Azerbaijan JBMO TST, 3
Find all the pime numbers $(p,q)$ such that :
$p^{3}+p=q^{2}+q$
2016-2017 SDML (Middle School), 1
What is the integer value of $\left(\sqrt{3}^{\sqrt{2}}\right)^{\sqrt{8}}$?
2007 Silk Road, 4
The set of polynomials $f_1, f_2, \ldots, f_n$ with real coefficients is called [i]special [/i], if for any different $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial $\dfrac{2}{3}f_i + f_j + f_k$ has no real roots, but for any different $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root.
a) Give an example of a [i]special [/i] set of four polynomials whose sum is not a zero polynomial.
b) Is there a [i]special [/i] set of five polynomials?
2021 Portugal MO, 3
All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$. What is the sum of the smallest elements of all these sequences?
2023 South East Mathematical Olympiad, 7
The positive integer number $S$ is called a "[i]line number[/i]". if there is a positive integer $n$ and $2n$ positive integers $a_1$, $a_2$,...,$a_n$, $b_1$,$b_2$,...,$b_n$, such that $S = \sum^n_{i=1} a_ib_i$, $\sum^n_{i=1} (a_i^2-b_1^2)=1$, and $\sum^n_{i=1} (a_i+b_i)=2023$, find:
(1) The minimum value of [i]line numbers[/i].
(2)The maximum value of [i]line numbers[/i].
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 $p + q > n$.