Found problems: 85335
2007 Princeton University Math Competition, 5
$A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?
2009 Purple Comet Problems, 10
Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly half way between those two adjacent towers. Before time 0 there are no towers, but at time 0 the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time 10 seconds.
1972 IMO Longlists, 39
How many tangents to the curve $y = x^3-3x\:\: (y = x^3 + px)$ can be drawn from different points in the plane?
2016 Purple Comet Problems, 15
The real numbers $x$, $y$, and $z$ satisfy the system of equations
$$x^2 + 27 = -8y + 10z$$
$$y^2 + 196 = 18z + 13x$$
$$z^2 + 119 = -3x + 30y$$
Find $x + 3y + 5z$.
LMT Accuracy Rounds, 2021 F9
There exist some number of ordered triples of real numbers $(x,y,z)$ that satisfy the following system of equations:
\begin{align*}
x+y+2z &= 6\\
x^2+y^2+2z^2 &= 18\\
x^3+y^3+2z^3&=54
\end{align*}
Given that the sum of all possible positive values of $x$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$,$b$,$c$, and $d$ are positive integers, $c$ is squarefree, and $\gcd(a,b,d)=1$, find the value of $a+b+c+d$.
2016 JBMO Shortlist, 4
Find all triplets of integers $(a,b,c)$ such that the number
$$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$
is a power of $2016$.
(A power of $2016$ is an integer of form $2016^n$,where n is a non-negative integer.)
2000 Czech and Slovak Match, 4
Let $P(x)$ be a polynomial with integer coefficients. Prove that the polynomial $Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1$ has no integer roots.
2024/2025 TOURNAMENT OF TOWNS, P2
A squared ${20} \times {20}$ board is split into two-squared dominoes. Prove that some line contains the centers of at least ten of such dominoes.
Alexandr Yuran
2006 Moldova MO 11-12, 6
Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$.
Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.
2012 Romania Team Selection Test, 4
Prove that a finite simple planar graph has an orientation so that every vertex has out-degree at most 3.
2005 International Zhautykov Olympiad, 2
Let $ r$ be a real number such that the sequence $ (a_{n})_{n\geq 1}$ of positive real numbers satisfies the equation $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{m \plus{} 1} \leq r a_{m}$ for each positive integer $ m$. Prove that $ r \geq 4$.
2008 Irish Math Olympiad, 5
Suppose that $ x, y$ and $ z$ are positive real numbers such that $ xyz \ge 1$.
(a) Prove that
$ 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
(b) Prove that
$ (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$ $ \le 3(x \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
1980 VTRMC, 4
Let $P(x)$ be any polynomial of degree at most $3.$ It can be shown that there are numbers $x_1$ and $x_2$ such that $\textstyle\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2),$ where $x_1$ and $x_2$ are independent of the polynomial $P.$
(a) Show that $x_1=-x_2.$
(b) Find $x_1$ and $x_2.$
1951 Putnam, A2
In the plane, what is the locus of points of the sum of the squares of whose distances from $n$ fixed points is a constant? What restrictions, stated in geometric terms, must be put on the constant so that the locus is non-null?
2023 China Team Selection Test, P16
Let $\Gamma, \Gamma_1, \Gamma_2$ be mutually tangent circles. The three circles are also tangent to a line $l$. Let $\Gamma, \Gamma_1$ be tangent to each other at $B_1$, $\Gamma, \Gamma_2$ be tangent to each other at $B_2$, $\Gamma_1, \Gamma_2$ be tangent to each other at $C$. $\Gamma, \Gamma_1, \Gamma_2$ are tangent to $l$ at $A, A_1, A_2$ respectively, where $A$ is between $A_1,A_2$. Let $D_1 = A_1C \cap A_2B_2, D_2 = A_2C \cap A_1B_1$. Prove that $D_1D_2$ is parallel to $l$.
2024 Princeton University Math Competition, A3 / B5
Joseph chooses a permutation of the numbers $1, 2, 3, 4, 5, 6$ uniformly at random. Then, he goes through his permutation, and deletes the numbers which are not the maximum among each of the preceding numbers. For example, if he chooses the permutation $3, 2, 4, 5, 1, 6,$ then he deletes $2$ and $1,$ leaving him with $3, 4, 5, 6.$ The expected number of numbers remaining can be expressed as $m/n$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$
2007 South East Mathematical Olympiad, 2
$AB$ is the diameter of semicircle $O$. $C$,$D$ are two arbitrary points on semicircle $O$. Point $P$ lies on line $CD$ such that line $PB$ is tangent to semicircle $O$ at $B$. Line $PO$ intersects line $CA$, $AD$ at point $E$, $F$ respectively. Prove that $OE$=$OF$.
1988 AMC 8, 6
$ \frac{(.2)^{3}}{(.02)^{2}}= $
$ \text{(A)}\ .2\qquad\text{(B)}\ 2\qquad\text{(C)}\ 10\qquad\text{(D)}\ 15\qquad\text{(E)}\ 20 $
1999 China National Olympiad, 2
Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then \[f(x)\geq\lambda(x -a)^3\] for all $x\geq0$. Find the equality condition.
2016 IMO, 6
There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if $n$ is odd.
(b) Prove that Geoff can never fulfill his wish if $n$ is even.
2019 Olympic Revenge, 2
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$
2013 China National Olympiad, 3
Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \[ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td\] holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.
2023 Sharygin Geometry Olympiad, 9
It is known that the reflection of the orthocenter of a triangle $ABC$ about its circumcenter lies on $BC$. Let $A_1$ be the foot of the altitude from $A$. Prove that $A_1$ lies on the circle passing through the midpoints of the altitudes of $ABC$.
2024 USAMTS Problems, 2
A regular hexagon is placed on top of a unit circle such that one vertex coincides with the center of the circle, exactly two vertices lie on the circumference of the circle, and exactly one vertex lies outside of the circle. Determine the area of the hexagon.
2000 Romania Team Selection Test, 3
Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.