Olympiad problems

2009 Indonesia MO, 1

Tags: modular arithmetic , quadratic , algebra , polynomial , divisibility tests , number theory

Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that \[ 4n^6 \plus{} n^3 \plus{} 5\] is divisible by $ 7$.

2001 Korea - Final Round, 1

Tags: function , modular arithmetic , quadratic , relatively prime , number theory

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

2001 IMO Shortlist, 2

Tags: number theory , calculus , algebra , quadratic

Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.

PEN A Problems, 82

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

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

2017 Romanian Masters In Mathematics, 4

Tags: quadratic , symmetry , graph , function

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

2008 Harvard-MIT Mathematics Tournament, 3

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

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

1969 AMC 12/AHSME, 32

Tags: algebra , polynomial , function , quadratic , analytic geometry , graphing lines , slope

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: $\textbf{(A) }3\qquad \textbf{(B) }4\qquad \textbf{(C) }5\qquad \textbf{(D) }6\qquad \textbf{(E) }11$

2011 Morocco National Olympiad, 2

Tags: quadratic , algebra

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

2003 AMC 10, 5

Tags: vieta , quadratic , algebra , polynomial

Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$? $ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

2006 National Olympiad First Round, 11

Tags: quadratic

What is the sum of the real roots of the equation $4x^4-3x^2+7x-3=0$? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ -3 \qquad\textbf{(D)}\ -4 \qquad\textbf{(E)}\ \text {None of above} $

2006 Czech and Slovak Olympiad III A, 2

Tags: quadratic , algebra

Let $m,n$ be positive integers such that the equation (in respect of $x$) \[(x+m)(x+n)=x+m+n\] has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.

2009 National Olympiad First Round, 19

Tags: quadratic

$ a$ is a real number. $ x_1$ and $ x_2$ are the distinct roots of $ x^2 \plus{} ax \plus{} 2 \equal{} x$. $ x_3$ and $ x_4$ are the distinct roots of $ (x \minus{} a)^2 \plus{} a(x \minus{} a) \plus{} 2 \equal{} x$. If $ x_3 \minus{} x_1 \equal{} 3(x_4 \minus{} x_2)$, then $ x_4 \minus{} x_2$ will be ? $\textbf{(A)}\ \frac {a}{2} \qquad\textbf{(B)}\ \frac {a}{3} \qquad\textbf{(C)}\ \frac {2a}{3} \qquad\textbf{(D)}\ \frac {3a}{2} \qquad\textbf{(E)}\ \text{None}$

1999 Baltic Way, 5

Tags: parabola , analytic geometry , slope , quadratic , algebra , conic

The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.

1964 Vietnam National Olympiad, 2

Tags: quadratic , algebra , parametric equation

Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$. Find the number of roots of the equation $x + | x^2 - 1 | = k$, where $k$ is a real constant.

2001 Saint Petersburg Mathematical Olympiad, 9.2

Tags: quadratic , algebra , quadratic trinomial , root

Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers? [I]Proposed by F. Petrov[/i]

1990 Iran MO (2nd round), 2

Tags: polynomial , quadratic , quadratic formula , algebra

Let $\alpha$ be a root of the equation $x^3-5x+3=0$ and let $f(x)$ be a polynomial with rational coefficients. Prove that if $f(\alpha)$ be the root of equation above, then $f(f(\alpha))$ is a root, too.

2000 Putnam, 2

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

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: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]

2002 Baltic Way, 16

Tags: greatest common divisor , quadratic , number theory

Find all nonnegative integers $m$ such that \[a_m=(2^{2m+1})^2+1 \] is divisible by at most two different primes.

2010 IMC, 4

Tags: function , abstract algebra , quadratic , set theory , number theory

Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.

2013 Stanford Mathematics Tournament, 7

Tags: algebra , polynomial , quadratic

Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.

2011 Mongolia Team Selection Test, 1

Tags: quadratic , modular arithmetic , ring theory , diophantine equation , number theory

Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

2014 Contests, 3

Tags: quadratic , polynomial , algebra

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

Fractal Edition 2, P2

Tags: quadratic , algebra

The real numbers $a$, $b$, and $c$ are such that the quadratic trinomials $ax^2 + bx + c$ and $cx^2 + bx + a$ each have two strictly positive real roots. Show that the sum of all these roots is at least $4$.

2005 Junior Balkan MO, 1

Tags: quadratic

Find all positive integers $x,y$ satisfying the equation \[ 9(x^2+y^2+1) + 2(3xy+2) = 2005 . \]

2013 Tuymaada Olympiad, 6

Tags: quadratic , absolute value , algebra

Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root. [i]K. Kokhas & F. Petrov[/i]