Found problems: 1148
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.
Find all values of $b$ such that the difference between the maximum and minimum values of $f(x) = x^2-2bx-1$ on the interval $[0, 1]$ is $1$.
Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.
Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$.
[i]Aaron Pixton.[/i]
There are two values of $ a$ for which the equation $ 4x^2 \plus{} ax \plus{} 8x \plus{} 9 \equal{} 0$ has only one solution for $ x$. What is the sum of these values of $ a$?
$ \textbf{(A)}\ \minus{}16\qquad
\textbf{(B)}\ \minus{}8\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 8\qquad
\textbf{(E)}\ 20$
The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.
Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots?
[i]N. Agakhanov, I. Bogdanov[/i]
Given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that $r=s$.
A circle of radius $ r$ is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is 1/2. What is $ r$?
$ \textbf{(A) } 2\sqrt {2} \plus{} 2\sqrt {3} \qquad \textbf{(B) } 3\sqrt {3} \plus{} \sqrt {2} \qquad \textbf{(C) } 2\sqrt {6} \plus{} \sqrt {3} \qquad \textbf{(D) } 3\sqrt {2} \plus{} \sqrt {6}\\
\textbf{(E) } 6\sqrt {2} \minus{} \sqrt {3}$
Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that
(i) $\frac{p+1}{2}$ is even but is not a power of $2$, and
(ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$.
[i]Proposed by Evan Chen[/i]
Find all pairs of real numbers $x$ and $y$ which satisfy the following equations:
\begin{align*}
x^2 + y^2 - 48x - 29y + 714 & = 0 \\
2xy - 29x - 48y + 756 & = 0
\end{align*}
For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
Let be given three quadratic polynomials:
$P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$.
Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.
Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.
In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.
The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if:
$ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$
Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true.
[i]Dan Schwarz[/i]
Prove:
(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$
(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$
Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.
Find all pairs $(a, b)$ of real numbers with the following property:
[list]Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.[/list]