Olympiad problems

2014 Contests, 3

Tags: quadratics , number theory , Divisors , Combinatorial Number Theory , EGMO , EGMO 2014

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

2007 Iran MO (3rd Round), 6

Tags: algebra , polynomial , quadratics , complex numbers , algebra proposed

Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$. [img]http://i14.tinypic.com/4tk6mnr.png[/img] a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.

2012 ITAMO, 4

Tags: modular arithmetic , quadratics , number theory proposed , number theory

Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation: \[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \] The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.

2011 Kazakhstan National Olympiad, 6

Tags: quadratics , integration , inequalities , algebra , algebra unsolved

Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$

2013 Brazil Team Selection Test, 1

Tags: quadratics , number theory , IMO Shortlist

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]

1992 AMC 12/AHSME, 28

Tags: quadratics , complex numbers , algebra , quadratic formula , AMC

Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is $ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $

1991 Arnold's Trivium, 84

Tags: quadratics , induction , vector , complex numbers

Find the number of positive and negative squares in the canonical form of the quadratic form $\sum_{i<j}(x_i-x_j)^2$ in $n$ variables. The same for the form $\sum_{i<j}x_i x_j$.

2019 PUMaC Algebra B, 5

Tags: algebra , quadratics , 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 $|\log_2(S)|$.

1991 Arnold's Trivium, 18

Tags: integration , linear algebra , matrix , quadratics , Gauss , calculus

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]

2016 Moldova Team Selection Test, 2

Tags: floor function , quadratics , function , pen

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

1992 India Regional Mathematical Olympiad, 6

Tags: inequalities , search , quadratics

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

1998 India National Olympiad, 5

Tags: quadratics , algebra , polynomial , inequalities , sum of roots , area of a triangle

Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

2020-IMOC, A3

Tags: quadratics , inequalities , IMOC , Taiwan

$\definecolor{A}{RGB}{250,120,0}\color{A}\fbox{A3.}$ Assume that $a, b, c$ are positive reals such that $a + b + c = 3$. Prove that $$\definecolor{A}{RGB}{200,0,200}\color{A} \frac{1}{8a^2-18a+11}+\frac{1}{8b^2-18b+11}+\frac{1}{8c^2-18c+11}\le 3.$$ [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1734[/color]

1990 IMO Longlists, 24

Tags: quadratics , inequalities , algebra , system of equations , algebra proposed

Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$: \[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]

2005 IMO Shortlist, 1

Tags: algebra , polynomial , roots , quadratics , coefficients , IMO Shortlist

Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

2002 Iran Team Selection Test, 12

Tags: quadratics , induction , group theory , combinatorics proposed , combinatorics

We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic. [i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.

1970 Polish MO Finals, 6

Tags: quadratics , algebra , inequalities

Find the smallest real number $A$ such that, for every quadratic polynomial $f(x)$ satisfying $ | f(x)| \le 1$ for $0 \le x \le 1$, it holds that $f' (0) \le A$.

2006 Nordic, 2

Tags: quadratics , algebra , algebra unsolved

Real numbers $x,y,z$ are not all equal and satisfy $x+\frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}=k$. Find all possible values of $k$.

2008 Harvard-MIT Mathematics Tournament, 16

Tags: analytic geometry , trigonometry , quadratics , geometry , circumcircle , algebra , quadratic formula

Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$.

2014-2015 SDML (High School), 3

Tags: quadratics , 2014-15 SDML Round 2 , SDML High School

Let $a$ and $b$ be the roots of the equation $x^2-47x+289=0$. Compute $\sqrt{a}+\sqrt{b}$.

1990 Brazil National Olympiad, 5

Tags: function , quadratics , algebra , domain , induction , linear algebra , matrix

Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$) Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$, show that $F_n(x)=x$ (for any valid x).

2010 Polish MO Finals, 2

Tags: modular arithmetic , quadratics , Online Math Open , trigonometry , number theory , number theory proposed

Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.

1999 Romania Team Selection Test, 3

Tags: quadratics , modular arithmetic , induction , floor function , binomial coefficients , algebra , special factorizations

Prove that for any positive integer $n$, the number \[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares. [i]Dorin Andrica[/i]

1977 AMC 12/AHSME, 23

Tags: quadratics , geometry , 3D geometry , Vieta , AMC

If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then $\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$ $\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$

PEN M Problems, 7

Tags: induction , quadratics , algebra , Recursive Sequences

Prove that the sequence $ \{y_{n}\}_{n \ge 1}$ defined by \[ y_{0}=1, \; y_{n+1}= \frac{1}{2}\left( 3y_{n}+\sqrt{5y_{n}^{2}-4}\right) \] consists only of integers.