Found problems: 85335
2014 India Regional Mathematical Olympiad, 1
Let $ABC$ be a triangle with $\angle ABC $ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$. Prove that $RX$ is perpendicular to $BC$.
2023 Durer Math Competition Finals, 4
Benedek wrote down the following numbers: $1$ piece of one, $2$ pieces of twos, $3$ pieces of threes, $... $, $50$ piecies of fifties. How many digits did Benedek write down?
2018 Saudi Arabia BMO TST, 2
Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.
2017 Brazil Undergrad MO, 1
A polynomial is called positivist if it can be written as a product of two non-constant polynomials with non-negative real coefficients. Let $f(x)$ be a polynomial of degree greater than one such that $f(x^n)$ is positivist for some positive integer $n$. Show that $f(x)$ is positivist.
2019 Yasinsky Geometry Olympiad, p2
The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD =
10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.
2010 Princeton University Math Competition, 5
Given that $x$, $y$, and $z$ are positive integers such that $\displaystyle{\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 2}$. Find the number of all possible $x$ values.
2020 LMT Fall, 24
In the Oxtingnle math team, there are $5$ students, numbered $1$ to $5$, all of which either always tell the truth or always lie. When Marpeh asks the team about how they did in a $10$ question competition, each student $i$ makes $5$ separate statements (so either they are all false or all true): "I got problems $i+1$ to $2i$, inclusive, wrong", and then "Student $j$ got both problems $i$ and $2i$ correct" for all $j \neq i$. What is the most problems the team could have gotten correctly?
[i]Proposed by Jeff Lin[/i]
2017 NIMO Summer Contest, 15
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where $N=6^{1999}$.
[i]Proposed by David Altizio[/i]
1965 IMO Shortlist, 3
Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.
1978 Bundeswettbewerb Mathematik, 1
Let $a, b, c$ be sides of a triangle. Prove that
$$\frac{1}{3} \leq \frac{a^2 +b^2 +c^2 }{(a+b+c)^2 } < \frac{1}{2}$$
and show that $\frac{1}{2}$ cannot be replaced with a smaller number.
2004 Brazil Team Selection Test, Problem 2
Let $(x+1)^p(x-3)^q=x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n$, where $p,q$ are positive integers.
(a) Prove that if $a_1=a_2$, then $3n$ is a perfect square.
(b) Prove that there exists infinitely many pairs $(p,q)$ for which $a_1=a_2$.
2018 BMT Spring, 14
Let $F_1 = 0$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$. Compute $$\sum_{n=1}^\infty \frac{\sum_{n=1}^\infty F_i}{3^n}.$$
2004 China Team Selection Test, 3
Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.
2010 Turkey MO (2nd round), 1
Let $A$ and $B$ be two points on the circle with diameter $[CD]$ and on the different sides of the line $CD.$ A circle $\Gamma$ passing through $C$ and $D$ intersects $[AC]$ different from the endpoints at $E$ and intersects $BC$ at $F.$ The line tangent to $\Gamma$ at $E$ intersects $BC$ at $P$ and $Q$ is a point on the circumcircle of the triangle $CEP$ different from $E$ and satisfying $|QP|=|EP|. \: AB \cap EF =\{R\}$ and $S$ is the midpoint of $[EQ].$ Prove that $DR$ is parallel to $PS.$
2012 Balkan MO, 2
Prove that
\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]
for all positive real numbers $x,y$ and $z$.
2014 CHMMC (Fall), 4
If $f(i, j, k) = f(i - 1, j + k , 2i - 1)$ and $f(0, j, k) = j + k$, evaluate $f(n, 0, 0)$.
2003 SNSB Admission, 1
Show that if a holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ has the property that the modulus of any of its derivatives (of any order) is everywhere dominated by $ 1, $ then $ |f(z)|\le e^{|\text{Im} (z)|} , $ for all complex numbers $ z. $
2006 Baltic Way, 3
Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x),\ldots , P_{m}(x)$ with real coefficients such that
\[P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}\]
2016 Taiwan TST Round 3, 2
Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions:
$(i)$ $a_0=a_n=1$;
$(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$;
$(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
$(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$.
Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.
2019 Kosovo National Mathematical Olympiad, 5
Let $ABCDE$ be a regular pentagon. Let point $F$ be intersection of segments $AC$ and $BD$. Let point $G$ be in segment $AD$ such that $2AD=3AG$. Let point $H$ be the midpoint of side $DE$. Show that the points $F,G,H$ lie on a line.
2002 Iran MO (3rd Round), 15
Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)
1985 Bulgaria National Olympiad, Problem 4
Seven points are given in space, no four of which are on a plane. Each of the segments with the endpoints in these points is painted black or red. Prove that there are two monochromatic triangles (not necessarily both of the same color) with no common edge. Does the statement hold for six points?
2008 Princeton University Math Competition, A2/B3
A [i]hypergraph[/i] consists of a set of vertices $V$ and a set of subsets of those vertices, each of which is called an edge. (Intuitively, it's a graph in which each edge can contain multiple vertices). Suppose that in some hypergraph, no two edges have exactly one vertex in common. Prove that one can color this hypergraph's vertices such that every edge contains both colors of vertices.
1956 AMC 12/AHSME, 37
On a map whose scale is $ 400$ miles to an inch and a half, a certain estate is represented by a rhombus having a $ 60^{\circ}$ angle. The diagonal opposite $ 60^{\circ}$ is $ \frac {3}{16}$ in. The area of the estate in square miles is:
$ \textbf{(A)}\ \frac {2500}{\sqrt {3}} \qquad\textbf{(B)}\ \frac {1250}{\sqrt {3}} \qquad\textbf{(C)}\ 1250 \qquad\textbf{(D)}\ \frac {5625\sqrt {3}}{2} \qquad\textbf{(E)}\ 1250\sqrt {3}$
2006 Pan African, 4
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.