Olympiad problems

2012 AMC 10, 17

Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2008 IMO, 3

Tags: prime divisor , quadratic , number theory

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

2003 China Western Mathematical Olympiad, 1

Tags: function , quadratic , calculus , induction , integration , algebra

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

2005 IberoAmerican, 1

Tags: quadratic , algebra

Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}

2011 Math Prize For Girls Problems, 18

Tags: relatively prime , quadratic , calculus , integration , polynomial , algebra , number theory

The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?

2024 All-Russian Olympiad Regional Round, 10.2

Tags: conic , quadratic , parabola , parallelogram , algebra , geometry

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

2018 Istmo Centroamericano MO, 4

Tags: number theory , trinomial , quadratic , perfect square

Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

2013 Germany Team Selection Test, 2

Tags: quadratic , number theory

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

2014 Greece Team Selection Test, 1

Tags: quadratic , induction , number theory , modular arithmetic

Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$ a) Prove that the sequence consists only of natural numbers. b) Check if there are terms of the sequence divisible by $2011$.

2013 F = Ma, 15

Tags: geometric transformation , quadratic , geometry , rotation

A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water? $\textbf{(A) } 0.25\\ \textbf{(B) } 0.33\\ \textbf{(C) } 0.5\\ \textbf{(D) } 0.67\\ \textbf{(E) } 0.75$

1997 Finnish National High School Mathematics Competition, 1

Tags: quadratic , product of roots , parameterization , logarithm , polynomial , algebra

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2004 AMC 12/AHSME, 15

Tags: quadratic , algebra , ratio

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $ 100$ meters. They next meet after Sally has run $ 150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 350 \qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 500$

2003 AIME Problems, 12

Tags: quadratic , pigeonhole principle , ceiling function

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?

1958 AMC 12/AHSME, 33

Tags: quadratic , vieta , quadratic formula , algebra

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$

2022 South Africa National Olympiad, 2

Tags: quadratic , system of equations , algebra

Find all pairs of real numbers $x$ and $y$ which satisfy the following equations: \begin{align*} x^2 + y^2 - 48x - 29y + 714 & = 0 \\ 2xy - 29x - 48y + 756 & = 0 \end{align*}

2012 ISI Entrance Examination, 4

Tags: quadratic , inequalities , polynomial , algebra

Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.

2010 N.N. Mihăileanu Individual, 1

Tags: conic , quadratic , algebra , polynomial

Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.

2005 AIME Problems, 6

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

Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.

2001 National Olympiad First Round, 11

Tags: quadratic , vieta , sfft

For how many integers $n$, does the equation system \[\begin{array}{rcl} 2x+3y &=& 7\\ 5x + ny &=& n^2 \end{array}\] have a solution over integers? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of the preceding} $

2004 National Olympiad First Round, 6

Tags: quadratic

For which of the following value of $n$, there exists integers $a,b$ such that $a^2 + ab-6b^2 = n$? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 19 \qquad\textbf{(C)}\ 29 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ 37 $

1998 USAMTS Problems, 2

Tags: quadratic , arithmetic series , quadratic formula , algebra

There are inﬁnitely many ordered pairs $(m,n)$ of positive integers for which the sum \[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\] is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.