Olympiad problems

2006 AMC 12/AHSME, 22

Tags: probability , trigonometry , quadratic , geometry , rectangle , similar triangles , trig identities

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}$

2015 India Regional MathematicaI Olympiad, 2

Tags: quadratic , algebra , polynomial

Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $ s \neq t$ such that $P(s) = t$ and $P(t) = s$. Prove that $b-st$ is a root of $x^2 + ax + b - st$.

2013 Indonesia MO, 6

Tags: quadratic , number theory

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 $2^m$.

2007 Junior Balkan Team Selection Tests - Moldova, 6

Tags: quadratic , trinomial , angle

The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$

2014 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , quadratic , geometry , geometric transformation , dilation , inequalities , complex number

Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.

1992 IMO Shortlist, 2

Tags: quadratic , algebra , functional equation , recurrence relation

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.

PEN P Problems, 15

Tags: quadratic , algebra , diophantine equation , additive number theory

Find all integers $m>1$ such that $m^3$ is a sum of $m$ squares of consecutive integers.

1997 All-Russian Olympiad, 1

Tags: quadratic , polynomial , inequalities , algebra

Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. [i]E. Malinnikova[/i]

2011 AIME Problems, 15

Tags: algebra , polynomial , probability , floor function , quadratic

Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.

1995 All-Russian Olympiad, 3

Tags: quadratic , polynomial , symmetry , function , complex number , algebra

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

2010 Contests, 1

Tags: quadratic , algebra , quadratic formula

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

2018 CMIMC Individual Finals, 3

Tags: quadratic

Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$.

2017 Thailand TSTST, 2

Tags: euler , function , modular arithmetic , quadratic , prime number , number theory

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2004 China Team Selection Test, 1

Tags: function , quadratic , 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)$.

2007 Tournament Of Towns, 4

Tags: polynomial , quadratic , algebra

Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?

1990 IMO Longlists, 24

Tags: quadratic , system of equations , algebra , inequalities

Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$: \[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]

1989 IMO Longlists, 24

Tags: geometry , quadratic , number theory

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.

PEN M Problems, 7

Tags: induction , quadratic , algebra , recursive sequences

Prove that the sequence $ \{y_{n}\}_{n \ge 1}$ defined by \[ y_{0}=1, \; y_{n+1}= \frac{1}{2}\left( 3y_{n}+\sqrt{5y_{n}^{2}-4}\right) \] consists only of integers.

1998 German National Olympiad, 4

Tags: polynomial , quadratic , algebra

Let $a$ be a positive real number. Then prove that the polynomial \[ p(x)=a^3x^3+a^2x^2+ax+a \] has integer roots if and only if $a=1$ and determine those roots.

2010 Greece National Olympiad, 1

Tags: quadratic , number theory

Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

2013 Today's Calculation Of Integral, 867

Tags: calculus , integration , quadratic , function , derivative , algebra , linear equation

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2008 ITest, 7

Tags: inequalities , quadratic , floor function

Find the number of integers $n$ for which $n^2+10n<2008$.

1987 IMO Longlists, 69

Tags: number theory , polynomial , prime number , quadratic

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

2013 Brazil Team Selection Test, 2

Tags: modular arithmetic , inequalities , quadratic , function , algebra , floor function

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

1991 India National Olympiad, 1

Tags: quadratic , modular arithmetic

Find the number of positive integers $n$ for which (i) $n \leq 1991$; (ii) 6 is a factor of $(n^2 + 3n +2)$.