Found problems: 85335
2005 Georgia Team Selection Test, 8
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.
2023 Sharygin Geometry Olympiad, 15
Let $ABCD$ be a convex quadrilateral. Points $X$ and $Y$ lie on the extensions beyond $D$ of the sides $CD$ and $AD$ respectively in such a way that $DX = AB$ and $DY = BC$. Similarly points $Z$ and $T$ lie on the extensions beyond $B$ of the sides $CB$ and $AB$ respectively in such a way that $BZ = AD$ and $BT = DC$. Let $M_1$ be the midpoint of $XY$, and $M_2$ be the midpoint of $ZT$. Prove that the lines $DM_1, BM_2$ and $AC$ concur.
2001 Greece National Olympiad, 4
The numbers $1$ to $500$ are written on a board. Two pupils $A$ and $B$ play the following game: A player in turn deletes one of the numbers from the board. The game is over when only two numbers remain. Player $B$ wins if the sum of the two remaining numbers is divisible by $3,$ otherwise $A$ wins. If $A$ plays first, show that $B$ has a winning strategy.
2020 Taiwan TST Round 1, 2
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
2007 Today's Calculation Of Integral, 172
Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$
1974 Poland - Second Round, 1
Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.
2009 Today's Calculation Of Integral, 463
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.
2016 ASDAN Math Tournament, 9
A cyclic quadrilateral $ABCD$ has side lengths $AB=14$, $BC=19$, $CD=26$, and $DA=29$. Compute the sine of the smaller angle between diagonals $AC$ and $BD$.
2014 China Northern MO, 8
Two people, $A$ and $B$, play the game of blowing up a balloon. The balloon will explode only when the volume of the balloon $V>2014$ mL. $A$ blows in $1$ mL first, and then they takes turns blowing. It is agreed that the gas blown by each person must not be less than the gas blown by the other party last time and should not be more than twice the amount of gas the other party blew last time. The agreement is that the person who blows up the balloon loses. Who has a winning strategy ? Briefly explain it. (Do not consider the change in volume caused by the change in tension when the balloon is inflated).
2016 Taiwan TST Round 1, 1
Let $AB$ be a chord on a circle $O$, $M$ be the midpoint of the smaller arc $AB$. From a point $C$ outside the circle $O$ draws two tangents to the circle $O$ at the points $S$ and $T$. Suppose $MS$ intersects with $AB$ at the point $E$, $MT$ intersects with $AB$ at the point $F$. From $E,F$ draw a line perpendicular to $AB$ that intersects with $OS,OT$ at the points $X,Y$, respectively. Draw another line from $C$ which intersects with the circle $O$ at the points $P$ and $Q$. Let $R$ be the intersection point of $MP$ and $AB$. Finally, let $Z$ be the circumcenter of triangle $PQR$.
Prove that $X$,$Y$ and $Z$ are collinear.
Brazil L2 Finals (OBM) - geometry, 2007.1
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$.
[img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]
2010 Sharygin Geometry Olympiad, 8
Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.
2022 Harvard-MIT Mathematics Tournament, 6
Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.
1983 IMO Longlists, 5
Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational.
[b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$.
[b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?
2018 Istmo Centroamericano MO, 4
Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.
2013 Greece Team Selection Test, 4
Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two).
Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids".
Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".
2012 Saint Petersburg Mathematical Olympiad, 1
$\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}$
Prove that system hasn`t solutions if $a,b,c$ are different.
1958 Miklós Schweitzer, 1
[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]
1987 IMO Longlists, 9
In the set of $20$ elements $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z\}$ we have made a random sequence of $28$ throws. What is the probability that the sequence $CUBA \ JULY \ 1987$ appears in this order in the sequence already thrown?
2022 Stanford Mathematics Tournament, 8
For all positive integers $m>10^{2022}$, determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$.
2007 AMC 12/AHSME, 1
One ticket to a show costs $ \$20$ at full price. Susan buys $ 4$ tickets using a coupon that gives her a $ 25\%$ discount. Pam buys $ 5$ tickets using a coupon that gives her a $ 30\%$ discount. How many more dollars does Pam pay than Susan?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$
2012 AMC 8, 17
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?
$\textbf{(A)}\hspace{.05in}3 \qquad \textbf{(B)}\hspace{.05in}4 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}6 \qquad \textbf{(E)}\hspace{.05in}7 $
2021 Junior Balkan Team Selection Tests - Romania, P3
Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$
The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.
2024 HMNT, 8
Compute the unique real numbers $x<3$ such that $$\sqrt{(3-x)(4-x)}+\sqrt{(4-x)(6-x)}+\sqrt{(6-x)(3-x)}=x.$$
2019 Iran Team Selection Test, 6
$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that
$$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$
[i]Proposed by Navid Safaei[/i]