Found problems: 1132
2013 Sharygin Geometry Olympiad, 14
Let $M$, $N$ be the midpoints of diagonals $AC$, $BD$ of a right-angled trapezoid $ABCD$ ($\measuredangle A=\measuredangle D = 90^\circ$).
The circumcircles of triangles $ABN$, $CDM$ meet the line $BC$ in the points $Q$, $R$.
Prove that the distances from $Q$, $R$ to the midpoint of $MN$ are equal.
2004 India Regional Mathematical Olympiad, 3
Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$
Prove that
(A) $\lambda _n$ is an integer
(B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .
2014 Harvard-MIT Mathematics Tournament, 6
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.
2009 India National Olympiad, 6
Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that:
$ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.
2004 Croatia Team Selection Test, 1
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$
1960 AMC 12/AHSME, 25
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $
2014 NIMO Problems, 7
Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.
[i]Proposed by Lewis Chen[/i]
2024 All-Russian Olympiad Regional Round, 10.2
On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.
2000 National Olympiad First Round, 2
Discriminant of a second degree polynomial with integer coefficients cannot be
$ \textbf{(A)}\ 23
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 33
$
2014 Math Prize For Girls Problems, 17
Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that
\[
\frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x
\]
for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.
1971 Canada National Olympiad, 4
Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.
2012 ELMO Shortlist, 1
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
1991 AIME Problems, 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
2013 India Regional Mathematical Olympiad, 4
A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial
1963 Vietnam National Olympiad, 2
For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.
2008 AIME Problems, 13
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
1979 AMC 12/AHSME, 27
An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each
such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will
[i]not[/i] have distinct positive real roots?
$\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$
2014 Singapore Senior Math Olympiad, 24
Find the number of integers $x$ which satisfy the equation $(x^2-5x+5)^{x+5}=1$.
2002 Germany Team Selection Test, 1
Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying
\[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\
1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases}
\]
Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.
2012 Indonesia TST, 4
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
2000 India Regional Mathematical Olympiad, 7
Find all real values of $a$ such that $x^4 - 2ax^2 + x + a^2 -a = 0$ has all its roots real.
2010 ELMO Shortlist, 3
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
2008 AMC 12/AHSME, 24
Let $ A_0\equal{}(0,0)$. Distinct points $ A_1,A_2,\ldots$ lie on the $ x$-axis, and distinct points $ B_1,B_2,\ldots$ lie on the graph of $ y\equal{}\sqrt{x}$. For every positive integer $ n$, $ A_{n\minus{}1}B_nA_n$ is an equilateral triangle. What is the least $ n$ for which the length $ A_0A_n\ge100$?
$ \textbf{(A)}\ 13\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 19\qquad
\textbf{(E)}\ 21$
1999 Harvard-MIT Mathematics Tournament, 5
Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.
2000 AMC 10, 12
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
[asy]
unitsize(8);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
draw((8,1)--(11,1)--(11,2)--(8,2)--cycle);
draw((19,0)--(20,0)--(20,5)--(19,5)--cycle);
draw((18,1)--(21,1)--(21,4)--(18,4)--cycle);
draw((17,2)--(22,2)--(22,3)--(17,3)--cycle);
draw((32,0)--(33,0)--(33,7)--(32,7)--cycle);
draw((29,3)--(36,3)--(36,4)--(29,4)--cycle);
draw((31,1)--(34,1)--(34,6)--(31,6)--cycle);
draw((30,2)--(35,2)--(35,5)--(30,5)--cycle);
label("Figure",(0.5,-1),S);
label("$0$",(0.5,-2.5),S);
label("Figure",(9.5,-1),S);
label("$1$",(9.5,-2.5),S);
label("Figure",(19.5,-1),S);
label("$2$",(19.5,-2.5),S);
label("Figure",(32.5,-1),S);
label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$