Found problems: 85335
May Olympiad L2 - geometry, 2020.4
Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point in
bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$.
The Golden Digits 2024, P3
Let $ABC$ be an acute scalene triangle with orthocentre $H{}$ and circumcentre $O.{}$ Let $P{}$ be an arbitrary point on the segment $OH$ and $O_a$ be the circumcentre of $PBC.{}$ The line $PO_a$ intersects the line $HA$ at $X_a.{}$ Define $X_b$ and $X_c$ similarly. Let $Q{}$ be the isogonal conjugate of $P{}$ and $X{}$ be the circumcentre of $X_aX_bX_c.{}$ Prove that $PQ$ and $HX$ are parallel.
[i]Proposed by David Anghel[/i]
2016 Korea National Olympiad, 2
A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$.
Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$.
Prove that $IE \cdot IK= IC \cdot IL$.
2007 Iran MO (3rd Round), 2
We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$, a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a \equal{} br\plus{}s$, and $ \Delta(s) <\Delta(b)$.
a) Prove that the following mapping is a degree mapping:
\[ \delta(n)\equal{}\mbox{Number of digits in the binary representation of }n\]
b) Prove that there exist a degree mapping $ \Delta_{0}$ such that for each degree mapping $ \Delta$ and for each $ n\neq0$, $ \Delta_{0}(n)\leq\Delta(n)$.
c) Prove that $ \delta \equal{}\Delta_{0}$
[img]http://i16.tinypic.com/4qntmd0.png[/img]
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. :)