Found problems: 85335
Denmark (Mohr) - geometry, 2009.1
In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$?
[img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]
2023 Azerbaijan National Mathematical Olympiad, 4
Solve the following diophantine equation in the set of nonnegative integers:
$11^{a}5^{b}-3^{c}2^{d}=1$.
2015 Belarus Team Selection Test, 2
The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$.
I. Gorodnin
2001 Baltic Way, 17
Let $n$ be a positive integer. Prove that at least $2^{n-1}+n$ numbers can be chosen from the set $\{1, 2, 3,\ldots ,2^n\}$ such that for any two different chosen numbers $x$ and $y$, $x+y$ is not a divisor of $x\cdot y$.
2022 Kosovo & Albania Mathematical Olympiad, 2
Let $ABC$ be an acute triangle. Let $D$ be a point on the line parallel to $AC$ that passes through $B$, such that $\angle BDC = 2\angle BAC$ as well as such that $ABDC$ is a convex quadrilateral. Show that $BD + DC = AC$.
2021 MMATHS, 12
$ABCD$ is a regular tetrahedron with side length 1. Points $X,$ $Y,$ and $Z,$ distinct from $A,$ $B,$ and $C,$ respectively, are drawn such that $BCDX,$ $ACDY,$ and $ABDZ$ are also regular tetrahedra. If the volume of the polyhedron with faces $ABC,$ $XYZ,$ $BXC,$ $XCY,$ $CYA,$ $YAZ,$ $AZB,$ and $ZBX$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd(a,c) = 1$ and $b$ squarefree, find $a+b+c.$
[i]Proposed by Jason Wang[/i]
2018 Taiwan TST Round 2, 2
Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation.
It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.
2017 ELMO Shortlist, 2
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
1978 IMO Shortlist, 7
We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$
2006 Stanford Mathematics Tournament, 7
Find all solutions to $aabb=n^4-6n^3$, where $a$ and $b$ are non-zero digits, and $n$ is an integer. ($a$ and $b$ are not necessarily distinct.)
1989 Brazil National Olympiad, 1
The sides of a triangle $T$, with vertices $(0,0)$,$(3,0)$ and $(0,3)$ are mirrors.
Show that one of the images of the triagle $T_1$ with vertices $(0,0)$,$(0,1)$ and $(2,0)$ is the triangle with vertices $(24,36)$,$(24,37)$ and $(26,36)$.
PEN O Problems, 49
Consider the set of all five-digit numbers whose decimal representation is a permutation of the digits $1, 2, 3, 4, 5$. Prove that this set can be divided into two groups, in such a way that the sum of the squares of the numbers in each group is the same.
Kharkiv City MO Seniors - geometry, 2013.11.4
In the triangle $ABC$, the heights $AA_1$ and $BB_1$ are drawn. On the side $AB$, points $M$ and $K$ are chosen so that $B_1K\parallel BC$ and $A_1 M\parallel AC$. Prove that the angle $AA_1K$ is equal to the angle $BB_1M$.
2011 Postal Coaching, 1
Let $X$ be the set of all positive real numbers. Find all functions $f : X \longrightarrow X$ such that
\[f (x + y) \ge f (x) + yf (f (x))\]
for all $x$ and $y$ in $X$.
2016 CCA Math Bonanza, I6
Let $a,b,c$ be non-zero real numbers. The lines $ax + by = c$ and $bx + cy = a$ are perpendicular and intersect at a point $P$ such that $P$ also lies on the line $y=2x$. Compute the coordinates of point $P$.
[i]2016 CCA Math Bonanza Individual #6[/i]
2021 Polish MO Finals, 1
Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ($p_1=2$, $p_2=3$, $p_3=3$ etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$. Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$.
[hide=example]For $k=2$ $p_1=2$, $p_2=3$, $N=6$. So in set $\{ 1,2,3,4,5,6 \}$ we can find $3$ number satisfying thesis: $2$, $3$ and $4$. ($1$ and $5$ are not divisible by $2$ or $3$, and $6$ is divisible by both of them so by even number of primes )[/hide]
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.