Found problems: 1148
Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.
After the schoolday is over, Juku must attend an extra math class. The teacher
writes a quadratic equation $ x^2\plus{} p_1x\plus{}q_1 \equal{} 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 \plus{} p_2x \plus{} q_2 \equal{} 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.
A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis.
Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.
Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.
An equilateral triangle has inside it a point with distances 5,12,13 from the vertices . Find its side.
Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.
Given any four distinct positive real numbers, show that one can choose three numbers $A,B,C$ from among them, such that all three quadratic equations \begin{eqnarray*} Bx^2 + x + C &=& 0\\ Cx^2 + x + A &=& 0 \\ Ax^2 + x +B &=& 0 \end{eqnarray*} have only real roots, or all three equations have only imaginary roots.
For what values of $a$ does the system of equations
\[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?
For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.
Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.
How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.
Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$.
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?
The teacher wrote on the board the quadratic polyomial $x^2+10x+20$. Then in turn, each of the students came to the board and increased or decreased by $1$ either the coefficient at $x$ or the constant term, but not both at once. As a result, the quadratic polyomial $x^2 + 20x +10$ appeared on the board. Is it true that at some point a quadratic polyomial with integer roots appeared on the board?
In the equation $ 2x^2 \minus{} hx \plus{} 2k \equal{} 0$, the sum of the roots is $ 4$ and the product of the roots is $ \minus{}3$. Then $ h$ and $ k$ have the values, respectively:
$ \textbf{(A)}\ 8\text{ and }{\minus{}6} \qquad
\textbf{(B)}\ 4\text{ and }{\minus{}3}\qquad
\textbf{(C)}\ {\minus{}3}\text{ and }4\qquad
\textbf{(D)}\ {\minus{}3}\text{ and }8\qquad
\textbf{(E)}\ 8\text{ and }{\minus{}3}$
Let $ a, b, c, d$ be positive integers such that $ ab \equal{} cd$ and $ a\plus{}b \equal{} c \minus{} d.$ Prove that there exists a right-angled triangle the measure of whose sides (in some unit) are integers and whose area measure is $ ab$ square units.
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
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$.
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Let $p=1601$. Prove that if
\[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},\]
where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$.
[i]Proposed by A. Golovanov[/i]
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]
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.
Suppose $a,b,c,A,B,C$ are real numbers with $a\ne0$ and $A\ne0$ such that for all $x$,
$$\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.$$Prove that
$$\left|b^2-4ac\right|\le\left|B^2-4AC\right|.$$
Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.