Olympiad problems

1995 India National Olympiad, 2

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

1970 Miklós Schweitzer, 4

Tags: modular arithmetic , quadratic , function , absolute value , number theory

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

1980 AMC 12/AHSME, 24

Tags: polynomial , quadratic , rational root theorem , algebra

For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$? $\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$

2009 Croatia Team Selection Test, 4

Tags: quadratic , diophantine equation , number theory

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

2002 All-Russian Olympiad, 1

Tags: polynomial , quadratic , ring theory , equation , functional equation , algebra

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

PEN P Problems, 31

Tags: quadratic , additive number theory

A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. [list=a] [*] Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$. [*] Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=1996$. [/list]

2009 USAMTS Problems, 2

Tags: quadratic , algebra , quadratic formula

Let $a, b, c, d$ be four real numbers such that \begin{align*}a + b + c + d &= 8, \\ ab + ac + ad + bc + bd + cd &= 12.\end{align*} Find the greatest possible value of $d$.

2016 Indonesia TST, 2

Tags: invariant , symmetry , quadratic , algebra , polynomial , vieta , number theory

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

1971 AMC 12/AHSME, 19

Tags: ellipse , quadratic , algebra , quadratic formula , conic

If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is $\textbf{(A) }\textstyle\frac{1}{2}\qquad\textbf{(B) }\frac{2}{3}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$

1979 AMC 12/AHSME, 27

Tags: probability , quadratic , search , absolute value

An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will [i]not[/i] have distinct positive real roots? $\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$

2002 AMC 12/AHSME, 23

Tags: quadratic , complex number , quadratic formula , polynomial , rational root theorem , algebra

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$. $\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$

1962 AMC 12/AHSME, 34

Tags: quadratic

For what real values of $ K$ does $ x \equal{} K^2 (x\minus{}1)(x\minus{}2)$ have real roots? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \minus{}2<K<1 \qquad \textbf{(C)}\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad \textbf{(D)}\ K>1 \text{ or } K<\minus{}2 \qquad \textbf{(E)}\ \text{all}$

2018 Istmo Centroamericano MO, 4

Tags: trinomial , quadratic , perfect square , number theory

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.

2002 Tournament Of Towns, 3

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

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

1999 India Regional Mathematical Olympiad, 7

Tags: quadratic , algebra , polynomial

Find the number of quadratic polynomials $ax^2 + bx +c$ which satisfy the following: (a) $a,b,c$ are distinct; (b) $a,b,c \in \{ 1,2,3,\cdots 1999 \}$; (c) $x+1$ divides $ax^2 + bx+c$.

2006 Taiwan National Olympiad, 3

Tags: quadratic , calculus , integration , algebra

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

Cono Sur Shortlist - geometry, 2012.G6.6

Tags: circumcircle , geometric transformation , reflection , quadratic , geometry

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

PEN A Problems, 18

Tags: quadratic , algebra , difference of squares , special factorizations , divisibility theory

Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

2010 AIME Problems, 10

Tags: quadratic , algebra , polynomial , counting , distinguishability

Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.

1995 IberoAmerican, 2

Tags: inequalities , quadratic , algebra

Let $n$ be a positive integer greater than 1. Determine all the collections of real numbers $x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0$ such that the next two conditions hold: (i) $x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12$ (ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}$

2002 Turkey MO (2nd round), 1

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

Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition \[y^2 \equiv x^3 - x \pmod p\] is exactly $p.$

1971 Bundeswettbewerb Mathematik, 4

Tags: quadratic , 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 $Q$.

2023 Auckland Mathematical Olympiad, 5

Tags: quadratic , combinatorics , algebra

There are $11$ quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this $$\star x^2 + \star x + \star= 0.$$ Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number. The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible. What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players.

2011 Kazakhstan National Olympiad, 6

Tags: quadratic , integration , inequalities , algebra

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 $2\sqrt{2n}\le a\le 3\sqrt{n}$

2012 ELMO Shortlist, 5

Tags: quadratic , modular arithmetic , 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]