Found problems: 85335
1990 IMO Longlists, 3
The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?
2003 All-Russian Olympiad Regional Round, 11.4
Points $ A_1,A_2,...,A_n$ and $ B_1,B_2,...,B_n$ are given on a plane. Show that the points $ B_i$ can be renumbered in such a way that the angle between vectors $ A_iA_j^{\longrightarrow}$ and $ B_iB_j^{\longrightarrow}$ is acute or right whenever $ i\neq j$.
2014 Oral Moscow Geometry Olympiad, 6
A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.
2015 USAMTS Problems, 3
For $n > 1$, let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$. Find the maximum value of $\frac{a_n}{n}$.
2020 Colombia National Olympiad, 2
Given a regular $n$-sided polygon with $n \ge 3$. Maria draws some of its diagonals in such a way that each diagonal intersects at most one of the other diagonals drawn in the interior of the polygon. Determine the maximum number of diagonals that Maria can draw in such a way.
Note: Two diagonals can share a vertex of the polygon. Vertices are not part of the interior of the polygon.
2010 Harvard-MIT Mathematics Tournament, 1
Suppose that $x$ and $y$ are positive reals such that \[x-y^2=3, \qquad x^2+y^4=13.\] Find $x$.
2013 HMNT, 8
How many of the fi rst $1000$ positive integers can be written as the sum of fi nitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?
2018-2019 Fall SDPC, 5
For a positive integer that doesn’t end in $0$, define its reverse to be the number formed by reversing its digits. For instance, the reverse of $102304$ is $403201$. In terms of $n \geq 1$, how many numbers when added to its reverse give $10^{n}-1$, the number consisting of $n$ nines?
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 $OE$=$OF$.