Olympiad problems

1955 AMC 12/AHSME, 27

Tags: quadratic

If $ r$ and $ s$ are the roots of $ x^2\minus{}px\plus{}q\equal{}0$, then $ r^2\plus{}s^2$ equals: $ \textbf{(A)}\ p^2\plus{}2q \qquad \textbf{(B)}\ p^2\minus{}2q \qquad \textbf{(C)}\ p^2\plus{}q^2 \qquad \textbf{(D)}\ p^2\minus{}q^2 \qquad \textbf{(E)}\ p^2$

2000 Moldova National Olympiad, Problem 1

Tags: algebra , quadratic

Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a $\frac a2x^2+bx+c=0$ between $r$ and $s$.

2002 AMC 10, 6

Tags: quadratic

For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{one} \qquad \textbf{(C)}\ \text{two} \qquad \textbf{(D)}\ \text{more than two, but finitely many}\\ \textbf{(E)}\ \text{infinitely many}$

2008 Vietnam National Olympiad, 6

Tags: inequalities , quadratic , algebra

Let $ x, y, z$ be distinct non-negative real numbers. Prove that \[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\] When does the equality hold?

1998 National Olympiad First Round, 4

Tags: function , quadratic

$ x,y,z\in \mathbb R$, find the minimal value of $ f\left(x,y,z\right) = 2x^{2} + 5y^{2} + 10z^{2} - 2xy - 4yz - 6zx + 3$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ -3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$

2010 Polish MO Finals, 2

Tags: modular arithmetic , quadratic , trigonometry , number theory

Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.

2002 Germany Team Selection Test, 1

Tags: quadratic , algebra , function

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

2008 Harvard-MIT Mathematics Tournament, 6

Tags: inequalities , conic , quadratic , parabola , analytic geometry , absolute value , function

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2017-2018 SDPC, 2

Tags: algebra , quadratic

Call a quadratic [i]invasive[/i] if it has $2$ distinct real roots. Let $P$ be a quadratic polynomial with real coefficients. Prove that $P(x)$ is invasive [b]if and only if[/b] there exists a real number $c \neq 0$ such that $P(x) + P(x - c)$ is invasive.

2011 USA TSTST, 6

Tags: inequalities , function , quadratic

Let $a, b, c$ be positive real numbers in the interval $[0, 1]$ with $a+b, b+c, c+a \ge 1$. Prove that \[ 1 \le (1-a)^2 + (1-b)^2 + (1-c)^2 + \frac{2\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}. \]

1951 AMC 12/AHSME, 47

Tags: quadratic

If $ r$ and $ s$ are the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$, the value of $ \frac {1}{r^2} \plus{} \frac {1}{s^2}$ is: $ \textbf{(A)}\ b^2 \minus{} 4ac \qquad\textbf{(B)}\ \frac {b^2 \minus{} 4ac}{2a} \qquad\textbf{(C)}\ \frac {b^2 \minus{} 4ac}{c^2} \qquad\textbf{(D)}\ \frac {b^2 \minus{} 2ac}{c^2}$ $ \textbf{(E)}\ \text{none of these}$

2016 Philippine MO, 1

Tags: algebra , invariant , discriminant , quadratic

The operations below can be applied on any expression of the form $ax^2+bx+c$. $(\text{I})$ If $c \neq 0$, replace $a$ by $4a-\frac{3}{c}$ and $c$ by $\frac{c}{4}$. $(\text{II})$ If $a \neq 0$, replace $a$ by $-\frac{a}{2}$ and $c$ by $-2c+\frac{3}{a}$. $(\text{III}_t)$ Replace $x$ by $x-t$, where $t$ is an integer. (Different values of $t$ can be used.) Is it possible to transform $x^2-x-6$ into each of the following by applying some sequence of the above operations? $(\text{a})$ $5x^2+5x-1$ $(\text{b})$ $x^2+6x+2$

1993 China Team Selection Test, 2

Tags: inequalities , algebra , function , quadratic

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

2008 Saint Petersburg Mathematical Olympiad, 5

Tags: modular arithmetic , quadratic , number theory

Given are distinct natural numbers $a$, $b$, and $c$. Prove that \[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]

2008 Czech-Polish-Slovak Match, 1

Tags: number theory , quadratic

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

2019 Paraguay Mathematical Olympiad, 1

Tags: algebra , trinomial , quadratic

Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem $-1$ and $-3$. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem $16$ and $-2$. What are the correct results of the problem?

2012 AIME Problems, 8

Tags: algebra , quadratic , complex number , quadratic formula

The complex numbers $z$ and $w$ satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of $|zw|^2$.

1959 AMC 12/AHSME, 34

Tags: vieta , irrational number , complex number , imaginary numbers , quadratic

Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

2014 NIMO Problems, 6

Tags: algebra , quadratic , derivative , quadratic formula , calculus

Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized? [i]Proposed by Lewis Chen[/i]

2003 China Team Selection Test, 2

Tags: combinatorics , quadratic , modular arithmetic

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

2002 AMC 10, 14

Tags: number theory , algebra , quadratic , vieta , prime number , special factorizations

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

2008 Harvard-MIT Mathematics Tournament, 6

Tags: trigonometry , polynomial , quadratic , hmmt , vieta , algebra

A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.

1994 Romania TST for IMO, 1:

Tags: number theory , quadratic , modular arithmetic

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

PEN D Problems, 11

Tags: symmetry , modular arithmetic , induction , quadratic , congruence

During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.

2021 Belarusian National Olympiad, 10.4

Tags: polynomial , quadratic , algebra

Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots. a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly. b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$