Found problems: 85335
2023 Hong Kong Team Selection Test, Problem 2
Giiven $\Delta ABC$, $\angle CAB=75^{\circ}$ and $\angle ACB=45^{\circ}$. $BC$ is extended to $T$ so that $BC=CT$. Let $M$ be the midpoint of the segment $AT$. Find $\angle BMC$.
1966 IMO Longlists, 55
Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$
2001 Macedonia National Olympiad, 1
Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$, then the equation $mx^2-sy^2=3$ has no integer solutions.
2016 CMIMC, 1
In a race, people rode either bicycles with blue wheels or tricycles with tan wheels. Given that 15 more people rode bicycles than tricycles and there were 15 more tan wheels than blue wheels, what is the total number of people who rode in the race?
1989 Irish Math Olympiad, 2
2. Each of $n$ members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.
Hi, from my sketches I'm thinking the answer is $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks
1968 AMC 12/AHSME, 28
If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $
1990 Baltic Way, 9
Two congruent triangles are inscribed in an ellipse. Are they necessarily symmetric with respect to an axis or the center of the ellipse?
2012 National Olympiad First Round, 27
What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$?
$ \textbf{(A)}\ \sqrt3 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ \sqrt 2 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$
2003 South africa National Olympiad, 3
The first four digits of a certain positive integer $n$ are $1137$. Prove that the digits of $n$ can be shuffled in such a way that the new number is divisible by 7.
2012 Bundeswettbewerb Mathematik, 3
An equilateral triangle $DCE$ is placed outside a square $ABCD$. The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ with $S$. Prove that the triangle $CMS$ is isosceles.
2018 HMNT, 4
Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.
2020 LIMIT Category 1, 5
Rohit is counting the minimum number of lines $m$, that can be drawn so that the number of distinct points of intersection exceeds $2020$. Find $m$.
(A)$63$
(B)$64$
(C)$65$
(D)$66$
2008 AMC 12/AHSME, 3
Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ \frac{5}{2} \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \frac{7}{2} \qquad
\textbf{(E)}\ 4$
2019 Saudi Arabia Pre-TST + Training Tests, 5.2
Let the bisector of the outside angle of $A$ of triangle $ABC$ and the circumcircle of triangle $ABC$ meet at point $P$. The circle passing through points $A$ and $P$ intersects segments $BP$ and $CP$ at points $E$ and $F$ respectively. Let $AD$ is the angle bisector of triangle $ABC$. Prove that $\angle PED = \angle PFD$.
[img]https://cdn.artofproblemsolving.com/attachments/0/3/0638429a220f07227703a682479ed150302aae.png[/img]
V Soros Olympiad 1998 - 99 (Russia), 8.1 - 8.4
[b]p1.[/b] Is it possible to write $5$ different fractions that add up to $1$, such that their numerators are equal to one and their denominators are natural numbers?
[b]p2.[/b] The following is known about two numbers $x$ and $y$:
if $x\ge 0$, then $y = 1 -x$;
if $y\le 1$, then $x = 1 + y$;
if $x\le 1$, then $x = |1 + y|$.
Find $x$ and $y$.
[b]p3.[/b] Five people living in different cities received a salary, some more, others less ($143$, $233$, $313$, $410$ and $413$ rubles). Each of them can send money to the other by mail. In this case, the post office takes $10\%$ of the amount of money sent for the transfer (in order to receive $100$ rubles, you need to send $10\%$ more, that is, $110$ rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method?
[b]p4.[/b] a) List three different natural numbers $m$, $n$ and $k$ for which $m! = n! \cdot k!$ .
b) Is it possible to come up with $1999$ such triplets?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
1997 Vietnam Team Selection Test, 3
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions:
1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;
2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.
1990 Romania Team Selection Test, 10
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
2010 ELMO Shortlist, 5
Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers.
[i]Brian Hamrick.[/i]
2011 Belarus Team Selection Test, 2
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$ Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$
I.Voronovich
LMT Team Rounds 2021+, A11 B17
In $\triangle ABC$ with $\angle BAC = 60^{\circ}$ and circumcircle $\omega$, the angle bisector of $\angle BAC$ intersects side $\overline{BC}$ at point $D$, and line $AD$ is extended past $D$ to a point $A'$. Let points $E$ and $F$ be the feet of the perpendiculars of $A'$ onto lines $AB$ and $AC$, respectively. Suppose that $\omega$ is tangent to line $EF$ at a point $P$ between $E$ and $F$ such that $\tfrac{EP}{FP} = \tfrac{1}{2}$. Given that $EF=6$, the area of $\triangle ABC$ can be written as $\tfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
[i]Proposed by Taiki Aiba[/i]
2020/2021 Tournament of Towns, P2
Maria has a balance scale that can indicate which of its pans is heavier or whether they have equal weight. She also has 4 weights that look the same but have masses of 1001, 1002, 1004 and 1005g. Can Maria determine the mass of each weight in 4 weightings? The weights for a new weighing may be picked when the result of the previous ones is known.
[i]The Jury[/i]
(For the senior paper) The same question when the left pan of the scale is lighter by 1g than the right one, so the scale indicates equality when the mass on the left pan is heavier by 1g than the mass on the right pan.
[i]Alexey Tolpygo[/i]
2009 VJIMC, Problem 1
A positive integer $m$ is called self-descriptive in base $b$, where $b\ge2$ is an integer, if
i) The representation of $m$ in base $b$ is of the form $(a_0a_1\ldots a_{b-1})_b$ (that is $m=a_0b^{b-1}+a_1b^{b-2}+\ldots+a_{b-2}b+a_{b-1}$, where $0\le a_i\le b-1$ are integers).
ii) $a_i$ is equal to the number of occurences of the number $i$ in the sequence $(a_0a_1\ldots a_{b-1})$.
For example, $(1210)_4$ is self-descriptive in base $4$, because it has four digits and contains one $0$, two $1$s, one $2$ and no $3$s.
2019 Balkan MO Shortlist, C1
100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.
1977 IMO Longlists, 3
In a company of $n$ persons, each person has no more than $d$ acquaintances, and in that company there exists a group of $k$ persons, $k\ge d$, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than $\left[ \frac{n^2}{4}\right]$.
2019 PUMaC Geometry A, 2
Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6, DB = 2$, and the ratio $\tfrac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5.$ Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.