Found problems: 1148
Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]
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 $
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.
Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to
$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$
What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac {1}{1000}$?
Given nonnegative real numbers $x_1,x_2,\dots ,x_n$, let $a=min\{x_1, x_2,\dots ,x_n\}$. Prove that the following inequality holds:
\[ \sum^{n}_{i=1}\dfrac{1+x_i}{1+x_{i+1}}\le n+\dfrac{1}{(1+a)^2}\sum^{n}_{i=1}(x_i-a)^2 \quad\quad (x_{n+1}=x_1),\]
and equality occurs if and only if $x_1=x_2=\dots =x_n$.
Consider the polynomials
\[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\]
Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$
If the function $f$ is defined by
\[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is
$\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]
Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$.
(1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin.
(2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized.
50 points
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of $f(81)$.
Find the remainder when when $N$ is divided by $p$.
[i]Proposed by Yang Liu and Ryan Alweiss[/i]
Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.
Prove that every polynomial of fourth degree can be represented in the form $P(Q(x))+R(S(x))$, where $P,Q,R,S$ are quadratic trinomials.
[i]A. Golovanov[/i]
[b]EDIT.[/b] It is confirmed that assuming the coefficients to be [b]real[/b], while solving the problem, earned a maximum score.
Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation:
\[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\]
Prove that there exists a unique solution of this equation.
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that:
\[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]
Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.
Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]
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?
Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.