Found problems: 85335
2023 HMNT, 30
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h,$ For every pair of people at the party, they are either friends or enemies. If every MIT student has $16$ MIT friends and $8$ Harvard friends, and every Harvard student has $7$ MIT enemies and $10$ Harvard enemies, compute how many pairs of friends there are at the party.
2023 Sharygin Geometry Olympiad, 24
A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.
2012 Kazakhstan National Olympiad, 1
Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?
2007 Stanford Mathematics Tournament, 9
Find $a^2+b^2$ given that $a, b$ are real and satisfy \[a=b+\frac{1}{a+\frac{1}{b+\frac{1}{a+\cdots}}}; b=a-\frac{1}{b+\frac{1}{a-\frac{1}{b+\cdots}}}\]
2012 NIMO Problems, 3
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
[i]Proposed by Aaron Lin[/i]
2018 MIG, 24
The sides of $\triangle ABC$ form an arithmetic sequence of integers. Incircle $I$ is tangent to $AB$, $BC$, and $CA$ at $D$, $E$, and $F$, respectively. Given that $DB = \tfrac32$, $FA = \tfrac12$, find the radius of $I$.
$\textbf{(A) } \dfrac12\qquad\textbf{(B) } \dfrac{\sqrt{15}}7\qquad\textbf{(C) } \dfrac{\sqrt{15}}6\qquad\textbf{(D) } \dfrac{2\sqrt{15}}{9}\qquad\textbf{(E) } \dfrac{\sqrt{15}}{4}$
2015 Harvard-MIT Mathematics Tournament, 2
The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$. Find $a+b$.
1987 Traian Lălescu, 1.2
On side $ BC $ of the triangle $ ABC, $ consider points $ M,N, $ such that $ MN=\frac{1}{n}BC. $ Let $ L $ on side $ AC $ such that $ ML $ is parallel with $ AB, Q $ on side $ AB $ such that $ NQ $ is parallel with $ AC, $ and $ O $ is the intersection of $ NQ $ with $ ML. $ The parallel of $ BC $ through $ O $ intersects $ AB,BC $ in $ P, $ respectively, $ K. $
Determine the location of $ M $ so that the sum of the areas of the triangles $ OMN, OKL $ and $ OPQ $ is minimum, and calculate this minimum in function of $ n $ and the area of $ ABC. $
2024 239 Open Mathematical Olympiad, 3
a) (version for grades 10-11)
Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle.
b) (version for grades 8-9)
Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?
1900 Eotvos Mathematical Competition, 1
Let $a, b, c, d$ be fixed integers with $d$ not divisible by $5$. Assume that $m$ is an integer for which $$am3 +bm2 +cm+d$$ is divisible by $5$. Prove that there exists an integer $n$ for which $$dn3 +cn2 +bn+a$$ is also divisible by $5$.
1980 Czech And Slovak Olympiad IIIA, 4
Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.
2022 Kazakhstan National Olympiad, 6
Numbers from $1$ to $49$ are randomly placed in a $35 \times 35$ table such that number $i$ is used exactly $i$ times. Some random cells of the table are removed so that table falls apart into several connected (by sides) polygons. Among them, the one with the largest area is chosen (if there are several of the same largest areas, a random one of them is chosen). What is the largest number of cells that can be removed that guarantees that in the chosen polygon there is a number which occurs at least $15$ times?
1998 All-Russian Olympiad, 4
A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.
2015 Math Prize for Girls Problems, 4
A [i]binary palindrome[/i] is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeroes are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?
2016 Saudi Arabia BMO TST, 3
Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying
$$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$
2006 AMC 10, 25
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2021 AMC 10 Spring, 24
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?
$\textbf{(A)}\ \frac{8a^2}{(a+1)^2}\qquad\textbf{(B)}\ \frac{4a}{a+1}\qquad\textbf{(C)}\ \frac{8a}{a+1}\qquad\textbf{(D)}\ \frac{8a^2}{a^2+1}\qquad\textbf{(E)}\ \frac{8a}{a^2+1}$
1997 Romania Team Selection Test, 1
We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.
[i]Mircea Becheanu[/i]
2018 MIG, 16
A triangle with area $60\text{ units}^2$ has vertices with coordinates of $(-15,x)$, $(0,x)$, and $(25,0)$. Find the largest possible value of $x$.
$\textbf{(A) } {-}8\qquad\textbf{(B) } {-}4\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 16$
2023 Princeton University Math Competition, 2
2. Let $\Gamma_{1}$ and $\Gamma_{2}$ be externally tangent circles with radii $\frac{1}{2}$ and $\frac{1}{8}$, respectively. The line $\ell$ is a common external tangent to $\Gamma_{1}$ and $\Gamma_{2}$. For $n \geq 3$, we define $\Gamma_{n}$ as the smallest circle tangent to $\Gamma_{n-1}, \Gamma_{n-2}$, and $\ell$. The radius of $\Gamma_{10}$ can be expressed as $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a+b$.
1999 North Macedonia National Olympiad, 5
If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$
2019 LIMIT Category A, Problem 5
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$.
$\textbf{(A)}~\frac n2$
$\textbf{(B)}~n$
$\textbf{(C)}~n\pi$
$\textbf{(D)}~\frac{n\pi}2$
1970 Miklós Schweitzer, 8
Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2}
\;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\]
[i]P. Turan[/i]
2020 Taiwan TST Round 1, 1
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
2013 Saudi Arabia GMO TST, 1
An acute triangle $ABC$ is inscribed in circle $\omega$ centered at $O$. Line $BO$ and side $AC$ meet at $B_1$. Line $CO$ and side $AB$ meet at $C_1$. Line $B_1C_1$ meets circle $\omega$ at $P$ and $Q$. If $AP = AQ$, prove that $AB = AC$.