Found problems: 1132
2011 Middle European Mathematical Olympiad, 1
Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that
\[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]
2006 Lithuania Team Selection Test, 2
Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.
2012 Online Math Open Problems, 48
Suppose that \[\sum_{i=1}^{982} 7^{i^2}\] can be expressed in the form $983q + r$, where $q$ and $r$ are integers and $0 \leq r \leq 492$. Find $r$.
[i]Author: Alex Zhu[/i]
2005 International Zhautykov Olympiad, 1
Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.
2014 Contests, 1
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.
2008 Serbia National Math Olympiad, 1
Find all nonegative integers $ x,y,z$ such that $ 12^x\plus{}y^4\equal{}2008^z$
1966 AMC 12/AHSME, 37
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals:
$\text{(A)}\ \dfrac{5}{2}\qquad
\text{(B)}\ \frac{3}{2}\qquad
\text{(C)}\ \dfrac{4}{3}\qquad
\text{(D)}\ \dfrac{5}{4}\qquad
\text{(E)}\ \dfrac{3}{4}$
2016 Middle European Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
1990 Iran MO (2nd round), 2
Let $\alpha$ be a root of the equation $x^3-5x+3=0$ and let $f(x)$ be a polynomial with rational coefficients. Prove that if $f(\alpha)$ be the root of equation above, then $f(f(\alpha))$ is a root, too.
2016 KOSOVO TST, 3
Equations $x^2+ax+b=0$ and $x^2+px+q=0$ have a common root.Find quadratic equation roots of which are two other roots.
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
2012 ISI Entrance Examination, 4
Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.
2002 AMC 10, 11
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
2011 Math Prize For Girls Problems, 7
If $z$ is a complex number such that
\[
z + z^{-1} = \sqrt{3},
\]
what is the value of
\[
z^{2010} + z^{-2010} \, ?
\]
2008 Moldova Team Selection Test, 1
Determine a subset $ A\subset \mathbb{N}^*$ having $ 5$ different elements, so that the sum of the squares of its elements equals their product.
Do not simply post the subset, show how you found it.
1999 India National Olympiad, 3
Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]
1983 AIME Problems, 3
What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
2006 AMC 12/AHSME, 24
Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which
\[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34?
\]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$
2012 Baltic Way, 4
Prove that for infinitely many pairs $(a,b)$ of integers the equation
\[x^{2012} = ax + b\]
has among its solutions two distinct real numbers whose product is 1.
1970 AMC 12/AHSME, 14
Consider $x^2+px+q=0$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1$, then $p$ equals
$\textbf{(A) }\sqrt{4q+1}\qquad\textbf{(B) }q-1\qquad\textbf{(C) }-\sqrt{4q+1}\qquad\textbf{(D) }q+1\qquad \textbf{(E) }\sqrt{4q-1}$
2007 IMO Shortlist, 6
Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even.
[url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url]
[i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]
2012 AIME Problems, 8
The complex numbers $z$ and $w$ satisfy the system
\begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*}
Find the smallest possible value of $|zw|^2$.
2010 Contests, 525
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.