Found problems: 85335
1983 AIME Problems, 1
Let $x$, $y$, and $z$ all exceed 1 and let $w$ be a positive number such that \[\log_x w = 24,\quad \log_y w = 40 \quad\text{and}\quad \log_{xyz} w = 12.\] Find $\log_z w$.
1996 All-Russian Olympiad, 3
Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$.
[i]A. Kovaldji, V. Senderov[/i]
2021 Bundeswettbewerb Mathematik, 4
Consider a pyramid with a regular $n$-gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue.
Show that if $n=9$ then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if $n=8$.
2015 Thailand TSTST, 1
Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.
2016 Turkey Team Selection Test, 7
$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?
2012 Irish Math Olympiad, 1
Let $$C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$$ and let
$$S=\{4,5,9,14,23,37\}$$ Find two sets $A$ and $B$ with the properties
(a) $A\cap B=\emptyset$.
(b) $A\cup B=C$.
(c) The sum of two distinct elements of $A$ is not in $S$.
(d) The sum of two distinct elements of $B$ is not in $S$.
2022 Purple Comet Problems, 1
Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.
2016 Hong Kong TST, 2
Let $\Gamma$ be a circle and $AB$ be a diameter. Let $l$ be a line outside the circle, and is perpendicular to $AB$. Let $X$, $Y$ be two points on $l$. If $X'$, $Y'$ are two points on $l$ such that $AX$, $BX'$ intersect on $\Gamma$ and such that $AY$, $BY'$ intersect on $\Gamma$. Prove that the circumcircles of triangles $AXY$ and $AX'Y'$ intersect at a point on $\Gamma$ other than $A$, or the three circles are tangent at $A$.
2023 AMC 8, 24
Isosceles $\triangle$ $ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle$ $ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle$ $ABC$?
[asy]
size(12cm);
draw((5,10)--(5,6.7),dashed+gray+linewidth(.5));
draw((5,3)--(5,5.3),dashed+gray+linewidth(.5));
filldraw((1.5,3)--(8.5,3)--(10,0)--(0,0)--cycle,lightgray);
draw((0,0)--(10,0)--(5,10)--cycle,linewidth(1.3));
dot((0,0));
dot((5,10));
dot((10,0));
label(scale(.8)*"$11$", (5,6.5),S);
dot((17.5,0));
dot((27.5,0));
dot((22.5,10));
draw((22.5,1.3)--(22.5,0),dashed+gray+linewidth(.5));
draw((22.5,2.5)--(22.5,3.6),dashed+gray+linewidth(.5));
draw((17.5,0)--(27.5,0)--(22.5,10)--cycle,linewidth(1.3));
filldraw((19.3,3.6)--(25.7,3.6)--(22.5,10)--cycle,lightgray);
label(scale(.8)*"$5$", (22.5,1.9));
draw((5,10)--(22.5,10),dashed+gray+linewidth(.5));
draw((10,0)--(17.5,0),dashed+gray+linewidth(.5));
draw((13.75,4.3)--(13.75,0),dashed+gray+linewidth(.5));
draw((13.75,5.7)--(13.75,10),dashed+gray+linewidth(.5));
label(scale(.8)*"$h$", (13.75,5));
label(scale(.7)*"$A$", (0,0), S);
label(scale(.7)*"$C$", (10,0), S);
label(scale(.7)*"$B$", (5,10), N);
label(scale(.7)*"$A$", (17.5,0), S);
label(scale(.7)*"$C$", (27.5,0), S);
label(scale(.7)*"$B$", (22.5,10), N);
[/asy]
$\textbf{(A) } 14.6 \qquad \textbf{(B) } 14.8 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 15.2 \qquad \textbf{(E) } 15.4$
1979 IMO Shortlist, 12
Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ;
(ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$
(iii) $\bigcup_{X \in F} X = R$
2008 Spain Mathematical Olympiad, 2
Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.
2014 Ukraine Team Selection Test, 1
Given an integer $n \ge 2$ and a regular $2n$-polygon at each vertex of which sitting on an ant. At some points in time, each ant creeps into one of two adjacent peaks (some peaks may have several ants at a time). Through $k$ such operations, it turned out to be an arbitrary line connecting two different ones the vertices of a polygon with ants do not pass through its center. For given $n$ find the lowest possible value of $k$.
2017 Macedonia JBMO TST, 3
Let $x,y,z$ be positive reals such that $xyz=1$. Show that
$$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$
When does equality happen?
1971 Canada National Olympiad, 5
Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.
1950 Miklós Schweitzer, 7
Examine the behavior of the expression
$ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$
as $ n\rightarrow \infty$
2018 Online Math Open Problems, 27
Let $n=2^{2018}$ and let $S=\{1,2,\ldots,n\}$. For subsets $S_1,S_2,\ldots,S_n\subseteq S$, we call an ordered pair $(i,j)$ [i]murine[/i] if and only if $\{i,j\}$ is a subset of at least one of $S_i, S_j$. Then, a sequence of subsets $(S_1,\ldots, S_n)$ of $S$ is called [i]tasty[/i] if and only if:
1) For all $i$, $i\in S_i$.
2) For all $i$, $\displaystyle\bigcup_{j\in S_i} S_j=S_i$.
3) There do not exist pairwise distinct integers $a_1,a_2,\ldots,a_k$ with $k\ge 3$ such that for each $i$, $(a_i, a_{i+1})$ is murine, where indices are taken modulo $k$.
4) $n$ divides $1+|S_1|+|S_2|+\ldots+|S_n|$.
Find the largest integer $x$ such that $2^x$ divides the number of tasty sequences $(S_1,\ldots, S_n)$.
[i]Proposed by Vincent Huang and Brandon Wang
2024 Dutch IMO TST, 1
Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.
2024 Saint Petersburg Mathematical Olympiad, 3
The triangle $ABC$ is inscribed in a circle. Two ants crawl out of points $B$ and $C$ at the same time. They crawl along the arc $BC$ towards each other so that the product of the distances from them to point $A$ remains unchanged. Prove that during their movement (until the moment of meeting), the straight line passing through the ants touches some fixed circle.
2020 Iranian Geometry Olympiad, 4
Let $P$ be an arbitrary point in the interior of triangle $\triangle ABC$. Lines$\overline{BP}$ and $\overline{CP}$
intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $\overline{CF}$ and $\overline{BE}$ intersect $\overline{BC}$ at $S$ and $T$, respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$, respectively. Prove that as $P$ moves in the interior of triangle $\triangle ABC$, line $\overline{MN}$ passes through a fixed point.
[i]Proposed by Ali Zamani[/i]
2022 CIIM, 3
Danielle draws a point $O$ on the plane and a set of points $\mathcal P = \{P_0, P_1, \ldots , P_{2022}\}$ such that $$\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,$$where the angles are measured counterclockwise and for $0 \leq n \leq 2022$, $OP_n = r^n$, where $r > 1$ is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points $\{A_0, A_1, \ldots , A_n\}$ in the plane, it is built a new set of points $\{B_0, B_1, \ldots , B_{n-1}\}$ such that $A_kA_{k+1}B_k$ is an equilateral triangle oriented clockwise for $0 \leq k \leq n - 1$. After carrying out the process $2022$ times from the set $P$, Danielle obtains a single point $X$. If the distance from $X$ to point $O$ is $d$, show that $$(r-1)^{2022} \leq d \leq (r+1)^{2022}.$$
2001 Canada National Olympiad, 5
Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$.
(1) Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear.
(2) Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.
2017 Danube Mathematical Olympiad, 2
Let n be a positive interger. Let n real numbers be wrote on a paper. We call a "transformation" :choosing 2 numbers $a,b$ and replace both of them with $a*b$. Find all n for which after a finite number of transformations and any n real numbers, we can have the same number written n times on the paper.
2010 District Olympiad, 4
Find all non negative integers $(a, b)$ such that
$$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$
2019 CCA Math Bonanza, T4
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\]
[i]2019 CCA Math Bonanza Team Round #4[/i]
2020 Latvia Baltic Way TST, 1
Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds:
$$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$