Olympiad problems

2003 Purple Comet Problems, 16

Find the largest real number $x$ such that \[\left(\dfrac{x}{x-1}\right)^2+\left(\dfrac{x}{x+1}\right)^2=\dfrac{325}{144}.\]

2004 Iran Team Selection Test, 1

Tags: modular arithmetic , quadratic , special factorizations , number theory

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2006 District Olympiad, 4

Tags: quadratic , abstract algebra , vector , group theory , algebra

a) Find two sets $X,Y$ such that $X\cap Y =\emptyset$, $X\cup Y = \mathbb Q^{\star}_{+}$ and $Y = \{a\cdot b \mid a,b \in X \}$. b) Find two sets $U,V$ such that $U\cap V =\emptyset$, $U\cup V = \mathbb R$ and $V = \{x+y \mid x,y \in U \}$.

2011 AMC 12/AHSME, 21

Tags: quadratic , algebra , difference of squares , special factorizations

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

1986 AMC 12/AHSME, 13

Tags: quadratic , parabola , conic

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$

2002 Iran MO (3rd Round), 16

Tags: quadratic , inequality , algebra

For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$

2000 AIME Problems, 6

Tags: quadratic , algebra , quadratic formula

For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$

1999 Romania National Olympiad, 3

Tags: complex number , quadratic , algebra

Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$ satisfy $|z_3|=|z_4|=1$

1994 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratic

Let be given three quadratic polynomials: $P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$. Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.

2005 Czech-Polish-Slovak Match, 3

Tags: polynomial , modular arithmetic , quadratic , algebra

Find all integers $n \ge 3$ for which the polynomial \[W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6\] can be written as a product of two non-constant polynomials with integer coefficients.

2001 Pan African, 1

Tags: search , quadratic

Find all positive integers $n$ such that: \[ \dfrac{n^3+3}{n^2+7} \] is a positive integer.

2012 Belarus Team Selection Test, 2

Tags: circumcircle , polynomial , quadratic , complex number , geometry , algebra

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

1980 AMC 12/AHSME, 8

Tags: quadratic , algebra , quadratic formula

How many pairs $(a,b)$ of non-zero real numbers satisfy the equation \[ \frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}? \] $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0$ $\text{(E)} \ \text{two pairs for each} ~b \neq 0$

PEN G Problems, 29

Tags: algebra , polynomial , quadratic , irrational number

Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?

2000 Korea - Final Round, 1

Tags: floor function , modular arithmetic , quadratic , number theory

Let $p$ be a prime such that $p \equiv 1 (\text {mod}4)$. Evaluate \[\sum_{k=1}^{p-1} \left( \left \lfloor \frac{2k^2}{p}\right \rfloor - 2 \left \lfloor {\frac{k^2}{p}}\right \rfloor \right)\]

2005 Polish MO Finals, 2

Tags: quadratic , number theory , sequence , relatively prime

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

2012 Stanford Mathematics Tournament, 8

Tags: quadratic

For real numbers $(x, y, z)$ satisfying the following equations, ﬁnd all possible values of $x+y+z$ $x^2y+y^2z+z^2x=-1$ $xy^2+yz^2+zx^2=5$ $xyz=-2$

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}$

2015 AMC 10, 23

Tags: quadratic , function , polynomial , vieta , sfft , algebra

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

1990 India National Olympiad, 2

Tags: calculus , integration , quadratic , number theory , diophantine equation

Determine all non-negative integral pairs $ (x, y)$ for which \[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]

2009 AMC 12/AHSME, 17

Tags: quadratic , geometric series

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$

2014 India IMO Training Camp, 1

Tags: quadratic , quadratic formula , algebra

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

2015 AMC 10, 12

Tags: quadratic , algebra , polynomial , vieta , quadratic formula , absolute value

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

2007 Bulgarian Autumn Math Competition, Problem 10.1

Tags: quadratic , algebra , roots of the equation

Find all integers $b$ and $c$ for which the equation $x^2-bx+c=0$ has two real roots $x_{1}$ and $x_{2}$ satisfying $x_{1}^2+x_{2}^2=5$.

1989 IMO Longlists, 83

Tags: number theory , diophantine equation , quadratic , equation

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]