Olympiad problems

2000 AIME Problems, 6

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 $y?$

2006 Estonia Team Selection Test, 1

Tags: algebra , quadratic , trinomial , integer polynomial , integer

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.

PEN A Problems, 82

Tags: function , quadratic , vieta , divisibility theory , vieta jumping , inequalities

Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

1986 USAMO, 3

Tags: relatively prime , number theory , algebra , quadratic

What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer? $\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]

1996 Chile National Olympiad, 4

Tags: quadratic , trinomial , algebra

Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$. Prove that $a\ge 5$ and $b\ge 5$.

2015 AMC 10, 14

Tags: quadratic

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$? $\textbf{(A) } 15 \qquad\textbf{(B) } 15.5 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 16.5 \qquad\textbf{(E) } 17 $

2012 NIMO Summer Contest, 9

Tags: quadratic , polynomial , algebra

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

1993 All-Russian Olympiad, 1

Tags: modular arithmetic , parameterization , quadratic , number theory

For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.

1991 India National Olympiad, 1

Tags: modular arithmetic , quadratic

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

1990 Czech and Slovak Olympiad III A, 2

Tags: triangle inequality , parameter , positive , quadratic , algebra , inequalities

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

2013 North Korea Team Selection Test, 3

Tags: modular arithmetic , number theory , quadratic , north korea , divisibility

Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

1992 AIME Problems, 13

Tags: inequalities , function , calculus , quadratic , analytic geometry , geometry , ratio

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

2010 AIME Problems, 13

Tags: relatively prime , number theory , probability , quadratic

The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a\plus{}9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\ge\frac12$ can be written as $ \frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

2011 ELMO Shortlist, 4

Tags: quadratic , modular arithmetic , 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]

2022 Indonesia TST, N

Tags: quadratic , greatest common divisor , diophantine , integer , number theory

For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

1998 Slovenia National Olympiad, Problem 2

Tags: number theory , quadratic

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.

1992 India Regional Mathematical Olympiad, 6

Tags: inequalities , search , quadratic

Prove that \[ 1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}. \]

2011 Federal Competition For Advanced Students, Part 1, 1

Tags: calculus , modular arithmetic , integration , quadratic , prime factorization , number theory

Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

1991 AIME Problems, 14

Tags: symmetry , geometry , circumcircle , quadratic , trigonometry

A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.

2019 PUMaC Algebra B, 5

Tags: algebra , quadratic , polynomial

Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

2005 AIME Problems, 15

Tags: incenter , geometry , quadratic , area of a triangle , heron's formula

Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$.

2014 Online Math Open Problems, 23

Tags: number theory , modular arithmetic , greatest common divisor , quadratic , law of cosines

For a prime $q$, let $\Phi_q(x)=x^{q-1}+x^{q-2}+\cdots+x+1$. Find the sum of all primes $p$ such that $3 \le p \le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying \[\dbinom{N}{\Phi_q(p)}\equiv \dbinom{2\Phi_q(p)}{N} \not \equiv 0 \pmod p. \][i]Proposed by Sammy Luo[/i]

PEN D Problems, 10

Tags: congruence , quadratic , modular arithmetic

Let $p$ be a prime number of the form $4k+1$. Suppose that $2p+1$ is prime. Show that there is no $k \in \mathbb{N}$ with $k<2p$ and $2^k \equiv 1 \; \pmod{2p+1}$.

2012 ELMO Shortlist, 5

Tags: modular arithmetic , quadratic , combinatorics

Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other. [i]Linus Hamilton.[/i]

2000 Putnam, 2

Tags: polynomial , algebra , quadratic , diophantine equation , induction

Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]