Found problems: 85335
2013 Greece JBMO TST, 2
Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle.
a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$
b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.
2016 Harvard-MIT Mathematics Tournament, 27
Find the smallest possible area of an ellipse passing through $(2,0)$, $(0,3)$, $(0,7)$, and $(6,0)$.
2004 China Team Selection Test, 1
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$
2006 Stanford Mathematics Tournament, 7
Let $S$ be the set of all 3-tuples $(a,b,c)$ that satisfy $a+b+c=3000$ and $a,b,c>0$. If one of these 3-tuples is chosen at random, what's the probability that $a,b$ or $c$ is greater than or equal to 2,500?
2000 Stanford Mathematics Tournament, 15
Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$?
2003 Tournament Of Towns, 7
A $m \times n$ table is filled with signs $"+"$ and $"-"$. A table is called irreducible if one cannot reduce it to the table filled with $"+"$, applying the following operations (as many times as one wishes).
$a)$ It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible $2\times 2$ sub table.
$b)$ It is allowed to flip all the signs in a row or in a column or on a diagonal (corner cells are diagonals of length $1$). Prove that an irreducible table contains an irreducible $4\times 4$ sub table.
2019 Online Math Open Problems, 6
An ant starts at the origin of the Cartesian coordinate plane. Each minute it moves randomly one unit in one of the directions up, down, left, or right, with all four directions being equally likely; its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Yannick Yao[/i]
KoMaL A Problems 2020/2021, A. 782
Prove that the edges of a simple planar graph can always be oriented such that the outdegree of all vertices is at most three.
[i]UK Competition Problem[/i]
1993 Poland - Second Round, 1
If $ x,y,u,v$ are positiv real numbers, prove the inequality :
\[ \frac {xu \plus{} xv \plus{} yu \plus{} yv}{x \plus{} y \plus{} u \plus{} v} \geq \frac {xy}{x \plus{} y} \plus{} \frac {uv}{u \plus{} v}
\]
2014 Harvard-MIT Mathematics Tournament, 19
Let $ABCD$ be a trapezoid with $AB\parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and $EDCF$ have areas $24$ and $36$, respectively, and triangle $ABH$ has area $25$. Find the area of triangle $CDG$.
Denmark (Mohr) - geometry, 1994.5
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment
2011 ELMO Shortlist, 3
Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples.
[i]Mitchell Lee.[/i]
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?