Olympiad problems

2015 AMC 12/AHSME, 20

Tags: geometry , perimeter , quadratics , area of a triangle , Heron's formula , algebra , quadratic formula

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

2010 All-Russian Olympiad Regional Round, 9.1

Tags: quadratics , algebra , polynomial , algebra proposed

Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at least one polynomial has two distinct roots.

1950 AMC 12/AHSME, 17

Tags: calculus , derivative , quadratics , algebra , system of equations , arithmetic series

The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is: \[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular}\] $\textbf{(A)}\ y=100-10x \qquad \textbf{(B)}\ y=100-5x^2 \qquad \textbf{(C)}\ y=100-5x-5x^2 \qquad\\ \textbf{(D)}\ y=20-x-x^2 \qquad \textbf{(E)}\ \text{None of these}$

2014 Contests, 2

Tags: quadratics , USA(J)MO , USAMO , induction , modular arithmetic , number theory proposed , Quadratic Residues

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

1986 AMC 12/AHSME, 13

Tags: quadratics , conics , parabola , AMC

A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$

1991 Romania Team Selection Test, 1

Tags: quadratics , number theory proposed , number theory

Suppose that $ a,b$ are positive integers for which $ A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a}$ is an integer.Prove that $ A\equal{}3$.

1987 AMC 12/AHSME, 23

Tags: inequalities , quadratics , AMC

If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then $ \textbf{(A)}\ 1<p\le 11 \qquad\textbf{(B)}\ 11<p \le 21 \qquad\textbf{(C)}\ 21< p \le 31 \\ \qquad\textbf{(D)}\ 31< p \le 41 \qquad\textbf{(E)}\ 41< p \le 51 $

1985 AMC 12/AHSME, 19

Tags: function , quadratics , special factorizations

Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect? $ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$ $ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$ $ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$

2016 Mathematical Talent Reward Programme, MCQ: P 13

Tags: quadratics , algebra

Let $P(x)=x^2+bx+c$. Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\neq P(-2)$. Then $P(0)=$ [list=1] [*] $-\frac{5}{2}$ [*] $-\frac{3}{2}$ [*] $-\frac{7}{4}$ [*] $\frac{6}{7}$ [/list]

1994 AIME Problems, 7

Tags: geometry , quadratics , analytic geometry , algebra , system of equations , linear equation

For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

2002 AMC 12/AHSME, 23

Tags: quadratics , complex numbers , algebra , quadratic formula , polynomial , Rational Root Theorem

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$. $\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$

2014 Purple Comet Problems, 23

Tags: quadratics

Suppose $x$ is a real number satisfying $x^2-990x+1=(x+1)\sqrt x$. Find $\sqrt x+\tfrac1{\sqrt x}$.

1986 IMO, 1

Tags: quadratics , modular arithmetic , number theory , system of equations , Perfect Squares , IMO , IMO 1986

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

1958 AMC 12/AHSME, 33

Tags: quadratics , Vieta , algebra , quadratic formula

For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows: $ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad \textbf{(B)}\ 2b^2 \equal{} 9ac\qquad \textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\ \textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad \textbf{(E)}\ 9b^2 \equal{} 2ac$

1962 AMC 12/AHSME, 34

Tags: quadratics

For what real values of $ K$ does $ x \equal{} K^2 (x\minus{}1)(x\minus{}2)$ have real roots? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \minus{}2<K<1 \qquad \textbf{(C)}\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad \textbf{(D)}\ K>1 \text{ or } K<\minus{}2 \qquad \textbf{(E)}\ \text{all}$

1990 IMO Longlists, 38

Tags: quadratics , algebra , number theory proposed , number theory

Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$

1998 Slovenia National Olympiad, Problem 2

Tags: quadratics , number theory

Find all pairs $(p,q)$ of real numbers such that $p+q=1998$ and the solutions of the equation $x^2+px+q=0$ are integers.

2006 Estonia Team Selection Test, 1

Tags: algebra , quadratics , trinomial , Integer Polynomial , Integers

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

2005 Junior Balkan Team Selection Tests - Moldova, 8

Tags: inequalities , trinomial , quadratics , quadratic polynomial , function , algebra

The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter. Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.

2017 All-Russian Olympiad, 2

Tags: quadratics , number theory , algebra , polynomial

$a,b,c$ - different natural numbers. Can we build quadratic polynomial $P(x)=kx^2+lx+m$, with $k,l,m$ are integer, $k>0$ that for some integer points it get values $a^3,b^3,c^3$ ?

2011 USA TSTST, 6

Tags: inequalities , USAMO , function , quadratics

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

2007 Mediterranean Mathematics Olympiad, 3

Tags: geometry , inradius , trigonometry , quadratics , geometry unsolved

In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

2000 Czech And Slovak Olympiad IIIA, 4

Tags: algebra , polynomial , quadratic polynomial , trinomial , quadratics

For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?

1999 Greece National Olympiad, 1

Tags: quadratics , inequalities proposed , inequalities

Let $f(x)=ax^2+bx+c$, where $a,b,c$ are nonnegative real numbers, not all equal to zero. Prove that $f(xy)^2\le f(x^2)f(y^2)$ for all real numbers $x,y$.

2011 ELMO Shortlist, 4

Tags: modular arithmetic , quadratics , number theory proposed , number theory

Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]