Found problems: 85335
1951 AMC 12/AHSME, 47
If $ r$ and $ s$ are the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$, the value of $ \frac {1}{r^2} \plus{} \frac {1}{s^2}$ is:
$ \textbf{(A)}\ b^2 \minus{} 4ac \qquad\textbf{(B)}\ \frac {b^2 \minus{} 4ac}{2a} \qquad\textbf{(C)}\ \frac {b^2 \minus{} 4ac}{c^2} \qquad\textbf{(D)}\ \frac {b^2 \minus{} 2ac}{c^2}$
$ \textbf{(E)}\ \text{none of these}$
2002 AMC 12/AHSME, 13
The sum of $ 18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is
$ \textbf{(A)}\ 169 \qquad
\textbf{(B)}\ 225 \qquad
\textbf{(C)}\ 289 \qquad
\textbf{(D)}\ 361 \qquad
\textbf{(E)}\ 441$
2015 Purple Comet Problems, 16
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made $60$ profit. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 profit. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars profit did Don make?
2014 ELMO Shortlist, 6
Let $ABCD$ be a cyclic quadrilateral with center $O$.
Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$.
Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$.
Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$.
Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$.
[i]Proposed by Yang Liu[/i]
1978 Putnam, B1
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and
the remaining four sides of length $2$. Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.
2012 India PRMO, 3
For how many pairs of positive integers $(x,y)$ is $x+3y=100$?
LMT Team Rounds 2021+, 2
How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive?
2021 Bangladesh Mathematical Olympiad, Problem 8
Shakur and Tiham are playing a game. Initially, Shakur picks a positive integer not greater than $1000$. Then Tiham picks a positive integer strictly smaller than that.Then they keep on doing this taking turns to pick progressively smaller and smaller positive integers until some one picks $1$. After that, all the numbers that have been picked so far are added up. The person picking the number $1$ wins if and only if this sum is a perfect square. Otherwise, the other player wins. What is the sum of all possible values of $n$ such that if Shakur starts with the number $n$, he has a winning strategy?
2011 Tournament of Towns, 4
Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?
1986 China National Olympiad, 1
We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$, then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.
1995 All-Russian Olympiad, 6
A boy goes $n$ times at a merry-go-round with $n$ seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which $n$ can he sit at every seat, if the lengths of all the $n-1$ moves he makes have different lengths?
[i]V. New[/i]
2019 Azerbaijan IMO TST, 1
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[ f(xy) = yf(x) + x + f(f(y) - f(x)) \]
for all $x,y \in \mathbb{R}$.
1989 Tournament Of Towns, (225) 3
A set of $1989$ numbers is given. It is known that the sum of any $10$ of them is positive. Prove that the sum of all these numbers is positive.
(Folklore)
2001 Kazakhstan National Olympiad, 1
Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.
2023 European Mathematical Cup, 4
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective.
[i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds.
[i]Ivan Novak[/i]
2023 IMAR Test, P1
Let $ABC$ be an acute triangle, and let $D,E,F$ be the feet of its altitudes from $A,B,C$ respectively. The lines $AB{}$ and $DE$ cross at $K{}$ and the lines $AC$ and $DF$ cross at $L{}.$ Let $M$ be the midpoint of the side $BC$ and let the line $AM$ cross the circle $(ABC)$ again at $N{}.$ The parallel through $M{}$ to $EF$ crosses the line $KL$ at $P{}.$ Prove that the triangle $MNP$ is isosceles.
2021 AMC 12/AHSME Fall, 3
Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?
$\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\
5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$
2020 Brazil Team Selection Test, 5
Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.
2019 AMC 12/AHSME, 16
There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a $\tfrac{1}{2}$ chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6?
$\textbf{(A) } \frac{15}{256} \qquad \textbf{(B) } \frac{1}{16} \qquad \textbf{(C) } \frac{15}{128}\qquad \textbf{(D) } \frac{1}{8} \qquad \textbf{(E) } \frac14$
2014 ELMO Shortlist, 3
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$.
[i]Proposed by Michael Kural[/i]
1995 All-Russian Olympiad Regional Round, 9.5
Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).
1974 IMO Longlists, 22
The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?
2007 Romania Team Selection Test, 1
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
2011 Morocco TST, 1
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
1985 Vietnam National Olympiad, 3
A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.