Found problems: 85335
1989 Spain Mathematical Olympiad, 4
Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.
2022 MIG, 2
Let $x$ be a number such that $10000x+2=4$. What is the value of $5000x+1$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }3$
2010 Purple Comet Problems, 9
What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$
2022 Turkey EGMO TST, 6
Let $x,y,z$ be positive real numbers satisfying the equations
$$xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0$$
What is the minimum value of the ratio of the sum of the largest and smallest numbers among $x,y,z$ to the median of them.
2023 Princeton University Math Competition, B1
Consider the equations $x^2+y^2=16$ and $xy=\tfrac{9}{2}.$ Find the sum, over all ordered pairs $(x,y)$ satisfying these equations, of $|x+y|.$
2007 Today's Calculation Of Integral, 204
Evaluate
\[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]
2023 USAMTS Problems, 3
Let $n \geq 2$ be a positive integer, and suppose buildings of height $1, 2, \ldots, n$ are built
in a row on a street. Two distinct buildings are said to be $\emph{roof-friendly}$ if every building
between the two is shorter than both buildings in the pair. For example, if the buildings are
arranged $5, 3, 6, 2, 1, 4,$ there are $8$ roof-friendly pairs: $(5, 3), (5, 6), (3, 6), (6, 2), (6, 4), (2, 1),$
$(2, 4), (1, 4).$ Find, with proof, the minimum and maximum possible number of roof-friendly
pairs of buildings, in terms of $n.$
Estonia Open Senior - geometry, 1995.2.4
Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.
1996 Romania Team Selection Test, 9
Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that
\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}
Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $.
1965 Putnam, B4
Consider the function
\[
f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots},
\]
where $n$ is a positive integer. Express $f(x,n+1)$ rationally in terms of $f(x,n)$ and $x$. Hence, or otherwise, evaluate $\textstyle\lim_{n\to\infty}f(x,n)$ for suitable fixed values of $x$. (The symbols $\textstyle\binom nr$ represent the binomial coefficients.)
2003 China Second Round Olympiad, 3
Let a space figure consist of $n$ vertices and $l$ lines connecting these vertices, with $n=q^2+q+1$, $l\ge q^2(q+1)^2+1$, $q\ge2$, $q\in\mathbb{N}$. Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least $p+2$ lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$ in the figure.
2001 AMC 10, 21
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $ 10$ and altitude $ 12$, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.
$ \textbf{(A)}\ \frac83 \qquad
\textbf{(B)}\ \frac{30}{11} \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \frac{25}{8} \qquad
\textbf{(E)}\ \frac{7}{2}$
2020 AMC 10, 7
How many positive even multiples of $3$ less than $2020$ are perfect squares?
$\textbf{(A) }7 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }10 \qquad\textbf{(E) }12$
2006 MOP Homework, 4
For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- [i]game [/i] defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.
2019 Teodor Topan, 2
Prove that a complex number $ z $ is real and positive if for any nonnegative integers $ n, $ the number
$$ z^{2^n} +\bar{z}^{2^n} $$
is real and positive.
[i]Sorin Rădulescu[/i]
1954 Moscow Mathematical Olympiad, 279
Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that
a) $OB< \frac{OA_1}{2}$ .
b) $OB \le \frac{OA_1}{4}$ .
[img]https://cdn.artofproblemsolving.com/attachments/5/f/5ea08453605e02e7e1253fd7c74065a9ffbd8e.png[/img]
1970 Canada National Olympiad, 9
Let $f(n)$ be the sum of the first $n$ terms of the sequence \[ 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . \] a) Give a formula for $f(n)$.
b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.
2010 Brazil Team Selection Test, 2
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2010 Bulgaria National Olympiad, 3
Let $k$ be the circumference of the triangle $ABC.$ The point $D$ is an arbitrary point on the segment $AB.$ Let $I$ and $J$ be the centers of the circles which are tangent to the side $AB,$ the segment $CD$ and the circle $k.$ We know that the points $A, B, I$ and $J$ are concyclic. The excircle of the triangle $ABC$ is tangent to the side $AB$ in the point $M.$ Prove that $M \equiv D.$
2017 Grand Duchy of Lithuania, 4
Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$.
(For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)
V Soros Olympiad 1998 - 99 (Russia), 10.4
Solve the equation $$ x + \sqrt{x^2-9} = \frac{2(x+3)}{(x-3)^2}$$
2001 AMC 12/AHSME, 15
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
$ \displaystyle \textbf{(A)} \ \frac {1}{2} \sqrt {3} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ \sqrt {2} \qquad \textbf{(D)} \ \frac {3}{2} \qquad \textbf{(E)} \ 2$
2003 India IMO Training Camp, 4
There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.
2011 India IMO Training Camp, 3
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
2023 Assam Mathematics Olympiad, 14
Find all possible triples of integers $a, b, c$ satisfying $a+b-c = 1$ and $a^2+b^2-c^2 =-1$.