Olympiad problems

2011 ELMO Shortlist, 4

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]

2005 Morocco TST, 1

Tags: quadratics , number theory , greatest common divisor , LaTeX , number theory unsolved

Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.

2000 AIME Problems, 6

Tags: quadratics , AMC , AIME , 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 $y?$

2018 Romania National Olympiad, 3

Tags: quadratics

Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer. Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$

1977 Canada National Olympiad, 6

Tags: quadratics , algebra , algebra proposed

Let $0 < u < 1$ and define \[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\] Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: function , quadratics , algebra , polynomial , arithmetic series , arithmetic sequence , quadratic formula

Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$. A. 1 B. 3 C. 7 D. 12 E. None of these

PEN A Problems, 4

Tags: search , algebra , polynomial , Vieta , symmetry , quadratics , Divisibility Theory

If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.

1989 IMO Longlists, 18

Tags: quadratics , combinatorics unsolved , combinatorics

There are some boys and girls sitting in an $ n \times n$ quadratic array. We know the number of girls in every column and row and every line parallel to the diagonals of the array. For which $ n$ is this information sufficient to determine the exact positions of the girls in the array? For which seats can we say for sure that a girl sits there or not?

2022 Indonesia TST, N

Tags: quadratics , greatest common divisor , number theory , Integers , diophantine

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 $n$.

1991 India National Olympiad, 1

Tags: quadratics , modular arithmetic

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

1998 Tournament Of Towns, 6

Tags: quadratics , function , composition , algebra

In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent: (i) there is a numerical interval without any values of $f(x)$ , (ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ . (A Kanel)

1971 Bundeswettbewerb Mathematik, 4

Tags: quadratics , modular arithmetic

Let $P$ and $Q$ be two horizontal neighbouring squares on a $n \times n$ chess board, $P$ on the left and $Q$ on the right. On the left square $P$ there is a stone that shall be moved around the board. The following moves are allowed: 1) move it one square upwards 2) move it one square to the right 3) move it one square down and one square to the left (diagonal movement) Example: you can get from $e5$ to $f5$, $e6$ and $d4$. Show that for no $n$ there is tour visting every square exactly once and ending in $Q$.

Oliforum Contest IV 2013, 4

Tags: algebra , polynomial , quadratics , number theory proposed , number theory , Contest 8

Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.

1994 Flanders Math Olympiad, 4

Tags: function , algebra , domain , quadratics

Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$) (a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined. (b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.

1963 Czech and Slovak Olympiad III A, 4

Tags: quadratic polynomial , Real Roots , roots , quadratics , algebra

Consider two quadratic equations \begin{align*}x^2+ax+b&=0, \\ x^2+cx+d&=0,\end{align*} with real coefficients. Find necessary and sufficient conditions such that the first equation has (real) roots $x,x_1,$ the second $x,x_2$ and $x>0,x_1>x_2$.

1952 AMC 12/AHSME, 6

Tags: quadratics

The difference of the roots of $ x^2 \minus{} 7x \minus{} 9 \equal{} 0$ is: $ \textbf{(A)}\ \plus{} 7 \qquad\textbf{(B)}\ \plus{} \frac {7}{2} \qquad\textbf{(C)}\ \plus{} 9 \qquad\textbf{(D)}\ 2\sqrt {85} \qquad\textbf{(E)}\ \sqrt {85}$

2013 JBMO TST - Turkey, 4

Tags: inequalities , quadratics , algebra , polynomial , inequalities proposed

For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that \[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]

1995 IMO Shortlist, 2

Tags: quadratics , modular arithmetic , number theory , partition , IMO Shortlist

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

2008 Czech-Polish-Slovak Match, 1

Tags: quadratics , number theory proposed , number theory

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

2012 AMC 12/AHSME, 10

Tags: geometry , analytic geometry , quadratics , conics , ellipse , AMC

What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$? ${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $

2004 IberoAmerican, 3

Tags: modular arithmetic , induction , quadratics , function , number theory unsolved , number theory

Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$

2004 USAMO, 3

Tags: geometry , ratio , function , calculus , quadratics , USAMO , Hi

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

2014 Contests, 4

Tags: quadratics , algebra , polynomial , calculus , integration , probability , analytic geometry

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

2011 Puerto Rico Team Selection Test, 2

Tags: quadratics , number theory , prime numbers , algebra

Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

1961 AMC 12/AHSME, 22

Tags: algebra , polynomial , quadratics , AMC

If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: ${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $