Olympiad problems

2009 Hungary-Israel Binational, 2

Tags: quadratic , inequalities

Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]

2004 China Team Selection Test, 1

Tags: quadratic , function , algebra

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)$.

1969 AMC 12/AHSME, 14

Tags: quadratic , inequalities

The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that: $\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\, \textbf{(B) }x>2\text{ or }x<-2$ $\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\, \textbf{(D) }x>1\text{ or }x<-2\qquad$ $\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$

2015 India PRMO, 2

Tags: polynomial , quadratic , algebra

$2.$ 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 $k ?$

2013 Hanoi Open Mathematics Competitions, 12

Tags: quadratic , root , inequalities , algebra

If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

2012 National Olympiad First Round, 19

Tags: quadratic , algebra

What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2014 Saudi Arabia BMO TST, 1

Tags: quadratic , algebra

Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers

1995 Cono Sur Olympiad, 3

Tags: geometry , quadratic , algebra , rectangle

Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$. 1. Find the distance between the centers of the circles(using $a$ and $b$). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

1999 Harvard-MIT Mathematics Tournament, 1

Tags: quadratic , function , algebra

If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?

2004 USAMO, 3

Tags: quadratic , calculus , function , ratio , geometry

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

1989 India National Olympiad, 1

Tags: calculus , quadratic , integration , modular arithmetic , polynomial , algebra

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

2012 Baltic Way, 18

Tags: quadratic , number theory

Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.

2011 Purple Comet Problems, 9

Tags: polynomial , quadratic , quadratic formula , sum of roots , product of roots , algebra

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

2002 Baltic Way, 16

Tags: greatest common divisor , quadratic , number theory

Find all nonnegative integers $m$ such that \[a_m=(2^{2m+1})^2+1 \] is divisible by at most two different primes.

2004 Junior Balkan MO, 3

Tags: modular arithmetic , number theory , quadratic

If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.

2000 National Olympiad First Round, 33

Tags: trigonometry , quadratic , pythagorean theorem , geometry , angle bisector

Let $K$ be a point on the side $[AB]$, and $L$ be a point on the side $[BC]$ of the square $ABCD$. If $|AK|=3$, $|KB|=2$, and the distance of $K$ to the line $DL$ is $3$, what is $|BL|:|LC|$? $ \textbf{(A)}\ \frac78 \qquad\textbf{(B)}\ \frac{\sqrt 3}2 \qquad\textbf{(C)}\ \frac 87 \qquad\textbf{(D)}\ \frac 38 \qquad\textbf{(E)}\ \frac{\sqrt 2}2 $

PEN A Problems, 1

Tags: number theory , quadratic , vieta , algebra , relatively prime , ratio

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

2012 Vietnam National Olympiad, 2

Tags: quadratic , arithmetic sequence , algebra , polynomial

Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.

1960 AMC 12/AHSME, 25

Tags: modular arithmetic , quadratic

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 $

2010 Math Prize For Girls Problems, 16

Tags: algebra , quadratic , analytic geometry , function , polynomial

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 $(a, b, c)$.