Found problems: 85335
1992 Romania Team Selection Test, 2
Let $ a_1, a_2, ..., a_k $ be distinct positive integers such that the $2^k$ sums $\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}$, $\epsilon_i\in\left\{0,1\right\}$ are distinct.
a) Show that $ \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k}) $;
b) Find the sequences $(a_1,a_2,...,a_k)$ for which the equality holds.
[i]Șerban Buzețeanu[/i]
1996 Argentina National Olympiad, 5
Determine all positive real numbers $x$ for which $$\left [x\right ]+\left [\sqrt{1996x}\right ]=1996$$ is verified
Clarification:The brackets indicate the integer part of the number they enclose.
2001 Baltic Way, 8
Let $ABCD$ be a convex quadrilateral, and let $N$ be the midpoint of $BC$. Suppose further that $\angle AND=135^{\circ}$.
Prove that $|AB|+|CD|+\frac{1}{\sqrt{2}}\cdot |BC|\ge |AD|.$
2011 Princeton University Math Competition, B1
If we define $\otimes(a,b,c)$ by
\begin{align*}
\otimes(a,b,c) = \frac{\max(a,b,c)- \min(a,b,c)}{a+b+c-\min(a,b,c)-\max(a,b,c)},
\end{align*}
compute $\otimes(\otimes(7,1,3),\otimes(-3,-4,2),1)$.
2022 Stanford Mathematics Tournament, 4
Let the roots of
\[x^{2022}-7x^{2021}+8x^2+4x+2\]
be $r_1,r_2,\dots,r_{2022}$, the roots of
\[x^{2022}-8x^{2021}+27x^2+9x+3\]
be $s_1,s_2,\dots,s_{2022}$, and the roots of
\[x^{2022}-9x^{2021}+64x^2+16x+4\]
be $t_1,t_2,\dots,t_{2022}$. Compute the value of
\[\sum_{1\le i,j\le2022}r_is_j+\sum_{1\le i,j\le2022}s_it_j+\sum_{1\le i,j\le2022}t_ir_j.\]
2016 Middle European Mathematical Olympiad, 8
For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers.
Prove that:
1. There does not exist a solution $(a, b, c)$ for $n = 2017$.
2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$.
3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.
2002 AMC 10, 24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $ 20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $ 10$ vertical feet above the bottom?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7.5 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 15$
2006 Austrian-Polish Competition, 4
A positive integer $d$ is called [i]nice[/i] iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ .
a) Is 29 nice?
b) Is 2006 nice?
c) Prove that infinitely many nice numbers exist.
2015 CHMMC (Fall), 2
You have $4$ game pieces, and you play a game against an intelligent opponent who has $6$. The rules go as follows: you distribute your pieces among two points a and b, and your opponent simultaneously does as well (so neither player sees what the other is doing). You win the round if you have more pieces than them on either $a$ or$ b$, and you lose the round if you only draw or have fewer pieces on both. You play the optimal strategy, assuming your opponent will play with the strategy that beats your strategy most frequently. What proportion of the time will you win?
2015 Iberoamerican Math Olympiad, 3
Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$:
$s_n = \alpha^n + \beta^n$
$t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$
Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.
1992 Flanders Math Olympiad, 4
Let $A,B,P$ positive reals with $P\le A+B$.
(a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that \[ A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)} \]
(b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$.
(c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$, $z,w>0$ and $z^2+w^2<x^2+y^2$.
Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$, the cyclical one has the greatest area.
2014 AMC 12/AHSME, 5
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$
2005 MOP Homework, 6
A positive integer $n$ is good if $n$ can be written as the sum of $2004$ positive integers $a_1$, $a_2$, ..., $a_{2004}$ such that $1 \le a_1 < a_2<...<a_{2004}$ and $a_i$ divides $a_{i+1}$ for $i=1$, $2$, ..., $2003$. Show that there are only finitely many positive integers that are not good.
2016 ISI Entrance Examination, 6
Suppose in a triangle $\triangle ABC$, $A$ , $B$ , $C$ are the three angles and $a$ , $b$ , $c$ are the lengths of the sides opposite to the angles respectively. Then prove that if $sin(A-B)= \frac{a}{a+b}\sin A \cos B - \frac{b}{a+b}\sin B \cos A$ then the triangle $\triangle ABC$ is isoscelos.
1999 Greece National Olympiad, 1
Let $f(x)=ax^2+bx+c$, where $a,b,c$ are nonnegative real numbers, not all equal to zero. Prove that $f(xy)^2\le f(x^2)f(y^2)$ for all real numbers $x,y$.
1974 IMO Longlists, 35
If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property:
If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$
2016 NIMO Problems, 3
Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area.
[i] Proposed by Michael Tang [/i]
1996 Austrian-Polish Competition, 3
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
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?
1952 AMC 12/AHSME, 24
In the figure, it is given that angle $ C \equal{} 90^{\circ}, \overline{AD} \equal{} \overline{DB}, DE \perp AB, \overline{AB} \equal{} 20$, and $ \overline{AC} \equal{} 12$. The area of quadrilateral $ ADEC$ is:
[asy]unitsize(7);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair A,B,C,D,E;
A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10);
draw(A--B--C--cycle); draw(D--E);
label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE);
draw(rightanglemark(B,D,E,30));[/asy]$ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 58\frac {1}{2} \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 37\frac {1}{2} \qquad\textbf{(E)}\ \text{none of these}$
2011 Tuymaada Olympiad, 3
Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called [i]singular[/i] if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there?
2004 Harvard-MIT Mathematics Tournament, 9
A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=a_n^2+1$ for each $n\ge0$. Find $\text{gcd}(a_{999},a_{2004})$.
2005 Czech And Slovak Olympiad III A, 3
In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$. Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ .
2010 Contests, 3
On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact.
A regular hexagon with its vertices on the circle is drawn on a circular billiard table.
A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges.
Describe a periodical track of this ball with exactly four points at the rails.
With how many different directions of impact can the ball be brought onto such a track?
2006 Purple Comet Problems, 6
We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?