Found problems: 1148
$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$
Show that \[ \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(\textbf{mod}\;p\ ) \] for every prime $ p\equiv 3 \;(\textbf{mod}\;4\ )$. [J. Suranyi]
A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.
Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that
[i](i)[/i] $ x$ and $ y$ are relatively prime;
[i](ii)[/i] $ y$ divides $ x^2 \plus{} m$;
[i](iii)[/i] $ x$ divides $ y^2 \plus{} m.$
[i](iv)[/i] $ x \plus{} y \leq m \plus{} 1\minus{}$ (optional condition)
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.
Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$
satisfy $|z_3|=|z_4|=1$
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.
Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate
and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.
Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$.
i) Prove there exist infinitely many primes, none dividing any term of the sequence.
ii) Prove there exist infinitely many primes, each dividing some term of the sequence.
[i](Dan Schwarz)[/i]
Suppose that $ f(x\plus{}3)\equal{}3x^2\plus{}7x\plus{}4$ and $ f(x)\equal{}ax^2\plus{}bx\plus{}c$. What is $ a\plus{}b\plus{}c$?
$ \textbf{(A)}\minus{}\!1 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is
$\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$
Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$?
$ \textbf{(A)}\ 19\qquad
\textbf{(B)}\ 22\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 27\qquad
\textbf{(E)}\ 32$
Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.
Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies
\[
\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},
\]
compute the ordered triple $(a, b, c)$.
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the
arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ ,
\[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]