Found problems: 85335
2021 Albanians Cup in Mathematics, 2
Angle bisector at $A$, altitude from $B$ to $CA$ and altitude of $C$ to $AB$ on a scalene triangle $ABC$ forms a triangle $\triangle$. Let $P$ and $Q$ points on lines $AB$ and $AC$, respectively, such that the midpoint of segment $PQ$ is the orthocenter of the triangle $\triangle$. Prove that the points $B, C, P$ and $Q$ lie on a circle.
1982 IMO Shortlist, 13
A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.
2010 Saint Petersburg Mathematical Olympiad, 5
$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$
2006 Estonia Team Selection Test, 5
Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2 \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$
2014 Harvard-MIT Mathematics Tournament, 26
For $1\leq j\leq 2014$, define \[b_j=j^{2014}\prod_{i=1, i\neq j}^{2014}(i^{2014}-j^{2014})\] where the product is over all $i\in\{1,\ldots,2014\}$ except $i=j$. Evaluate \[\dfrac1{b_1}+\dfrac1{b_2}+\cdots+\dfrac1{b_{2014}}.\]
1980 Putnam, A2
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that
$$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$
depending only on $r$ and $s.$
2015 Sharygin Geometry Olympiad, P22
The faces of an icosahedron are painted into $5$ colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.
1986 All Soviet Union Mathematical Olympiad, 435
All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules:
[i]At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column.
[/i]
a) What is the minimal
b) What is the maximal
possible number of $+1$ in the obtained table?
2000 CentroAmerican, 1
Find all three-digit numbers $ abc$ (with $ a \neq 0$) such that $ a^{2}+b^{2}+c^{2}$ is a divisor of 26.
VI Soros Olympiad 1999 - 2000 (Russia), 9.4
For real numbers $x \ge 0$ and $y \ge 0$, prove the inequality $$x^4+y^3+x^2+y+1 >\frac92 xy.$$
2008 ITest, 96
Triangle $ABC$ has $\angle A=90^\circ$, $\angle B=60^\circ$, and $AB=8$, and a point $P$ is chosen inside the triangle. The interior angle bisectors $\ell_A$, $\ell_B$, and $\ell_C$ of respective angles $PAB$, $PBC$, and $PCA$ intersect pairwise at $X=\ell_A\cap\ell_B$, $Y=\ell_B\cap\ell_C$, and $Z=\ell_C\cap\ell_A$. If triangles $ABC$ and $XYZ$ are directly similar, then the area of $\triangle XYZ$ may be written in the form $\tfrac{p\sqrt q-r\sqrt s}t$, where $p,q,r,s,t$ are positive integers, $q$ and $s$ are not divisible by the square of any prime, and $\gcd(t,r,p)=1$. Compute $p+q+r+s+t$.
Ukrainian TYM Qualifying - geometry, 2017.3
The altitude $AH, BT$, and $CR$ are drawn in the non isosceles triangle $ABC$. On the side $BC$ mark the point $P$; points $X$ and $Y$ are projections of $P$ on $AB$ and $AC$. Two common external tangents to the circumscribed circles of triangles $XBH$ and $HCY$ intersect at point $Q$. The lines $RT$ and $BC$ intersect at point $K$.
a). Prove that the point $Q$ lies on a fixed line independent of choice$ P$.
b). Prove that $KQ = QH$.
2022 Novosibirsk Oral Olympiad in Geometry, 5
Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]
2013 Stanford Mathematics Tournament, 5
A rhombus has area $36$ and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus?
2023 Turkey Olympic Revenge, 6
In triangle $ABC$, $D$ is a variable point on line $BC$. Points $E,F$ are on segments $AC, AB$ respectively such that $BF=BD$ and $CD=CE$. Circles $(AEF)$ and $(ABC)$ meet again at $S$. Lines $EF$ and $BC$ meet at $P$ and circles $(PDS)$ and $(AEF)$ meet again at $Q$. Prove that, as $D$ varies, isogonal conjugate of $Q$ with respect to triangle $ ABC$ lies on a fixed circle.
[i]Proposed by Serdar Bozdag[/i]
1992 APMO, 4
Determine all pairs $(h,s)$ of positive integers with the following property:
If one draws $h$ horizontal lines and another $s$ lines which satisfy
(i) they are not horizontal,
(ii) no two of them are parallel,
(iii) no three of the $h + s$ lines are concurrent,
then the number of regions formed by these $h + s$ lines is 1992.
2019 Math Prize for Girls Problems, 3
The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$.
2015 Online Math Open Problems, 4
Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_A$ through $O$ and $A$, and another circle $\omega_B$ through $O$ and $B$; the circles $\omega_A$ and $\omega_B$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega$, $\omega_A$, $\omega_B$ are congruent. If $CO = \sqrt 3$, what is the perimeter of $\triangle ABC$?
[i]Proposed by Evan Chen[/i]
2004 Iran Team Selection Test, 1
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that:
\[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]
1997 Greece Junior Math Olympiad, 3
Establish if we can rewrite the numbers $1,2,3,4,5,6,7,8,9,10$ in a row in such a way that:
(a) The sum of any three consecutive numbers (in the new order) does not exceed $16$.
(b) The sum of any three consecutive numbers (in the new order) does not exceed $15$.
2008 iTest Tournament of Champions, 4
If $m$ is a positive integer, let $S_m$ be the set of rational numbers in reduced form with denominator at most $m$. Let $f(m)$ be the sum of the numerator and denominator of the element of $S_m$ closest to $e$ (Euler's constant). Given that $f(2007) = 3722$, find the remainder when $f(1000)$ is divided by $2008$.
2011 Korea National Olympiad, 4
Let $k,n$ be positive integers. There are $kn$ points $P_1, P_2, \cdots, P_{kn}$ on a circle. We can color each points with one of color $ c_1, c_2, \cdots , c_k $. In how many ways we can color the points satisfying the following conditions?
(a) Each color is used $ n $ times.
(b) $ \forall i \not = j $, if $ P_a $ and $ P_b $ is colored with color $ c_i $ , and $ P_c $ and $ P_d $ is colored with color $ c_j $, then the segment $ P_a P_b $ and segment $ P_c P_d $ doesn't meet together.
2014 Romania National Olympiad, 4
Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $
[b]a)[/b] Prove that the order of $ G $ is a power of $ p. $
[b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $
LMT Speed Rounds, 4
The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even.
[i]Proposed by Muztaba Syed and Derek Zhao[/i]
[hide=Solution]
[i]Solution. [/i]$\boxed{\dfrac{1}{3}}$
Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the
answer is $\boxed{\dfrac{1}{3}}$.[/hide]
Bangladesh Mathematical Olympiad 2020 Final, #6
Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$[b]ABP =[/b] $\angle$[b]ADP[/b]. What is the radius of the circumcircle of $ABCD$?