Olympiad problems

2010 India Regional Mathematical Olympiad, 2

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

2005 AIME Problems, 12

Tags: analytic geometry , trigonometry , quadratics , geometry , geometric transformation , rotation , reflection

Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

1979 IMO Longlists, 42

Tags: quadratics , algebra , polynomial , algebra unsolved

Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$

1992 Spain Mathematical Olympiad, 3

Tags: quadratics , number theory

Prove that if $a,b,c,d$ are nonnegative integers satisfying $(a+b)^2+2a+b= (c+d)^2+2c+d$, then $a = c $ and $b = d$. Show that the same is true if $a,b,c,d$ satisfy $(a+b)^2+3a+b=(c+d)^2+3c+d$, but show that there exist $a,b,c,d $ with $a \ne c$ and $b \ne d$ satisfying $(a+b)^2+4a+b = (c+d)^2+4c+d$.

1969 IMO Longlists, 63

Tags: quadratics , number theory , Additive Number Theory , IMO Shortlist , IMO Longlist

$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

1959 IMO Shortlist, 3

Tags: quadratics , trigonometry , algebra , Trigonometric Equations , IMO , IMO 1959

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

2015 Romania National Olympiad, 2

Tags: function , algebra , polynomial , quadratics

A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $ Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $

2010 Stanford Mathematics Tournament, 4

Tags: ratio , quadratics , algebra , quadratic formula

Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$

2003 India Regional Mathematical Olympiad, 6

Tags: quadratics , algebra , polynomial

Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$.

2005 Morocco TST, 3

Tags: quadratics , calculus , integration , number theory unsolved , number theory

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

2009 Romania Team Selection Test, 3

Tags: quadratics , algebra , polynomial , search , number theory proposed , number theory

Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.

1985 IMO Longlists, 76

Tags: quadratics , number theory unsolved , number theory

Are there integers $m$ and $n$ such that \[5m^2 - 6mn + 7n^2 = 1985 \ ?\]

2008 Vietnam National Olympiad, 6

Tags: inequalities , quadratics , inequalities unsolved , algebra , High school olympiad , Vietnam

Let $ x, y, z$ be distinct non-negative real numbers. Prove that \[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\] When does the equality hold?

2008 JBMO Shortlist, 7

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

Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.

1977 Canada National Olympiad, 1

Tags: quadratics

If $f(x) = x^2 + x$, prove that the equation $4f(a) = f(b)$ has no solutions in positive integers $a$ and $b$.

1995 Iran MO (2nd round), 2

Tags: function , ceiling function , quadratics , number theory proposed , number theory

Let $n \geq 0$ be an integer. Prove that \[ \lceil \sqrt n +\sqrt{n+1}+\sqrt{n+2} \rceil = \lceil \sqrt{9n+8} \rceil\] Where $\lceil x \rceil $ is the smallest integer which is greater or equal to $x.$

2008 Saint Petersburg Mathematical Olympiad, 5

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

Given are distinct natural numbers $a$, $b$, and $c$. Prove that \[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]

2005 Germany Team Selection Test, 1

Tags: quadratics , number theory , Sequence , relatively prime , IMO Shortlist

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

2003 China Western Mathematical Olympiad, 1

Tags: calculus , integration , induction , function , quadratics , algebra unsolved , 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$.

2014 Iran Team Selection Test, 4

Tags: quadratics , number theory unsolved , number theory

$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

1957 AMC 12/AHSME, 39

Tags: quadratics , algebra , quadratic formula

Two men set out at the same time to walk towards each other from $ M$ and $ N$, $ 72$ miles apart. The first man walks at the rate of $ 4$ mph. The second man walks $ 2$ miles the first hour, $ 2\frac {1}{2}$ miles the second hour, $ 3$ miles the third hour, and so on in arithmetic progression. Then the men will meet: $ \textbf{(A)}\ \text{in 7 hours} \qquad \textbf{(B)}\ \text{in }{8\frac {1}{4}}\text{ hours}\qquad \textbf{(C)}\ \text{nearer }{M}\text{ than }{N}\qquad \\ \textbf{(D)}\ \text{nearer }{N}\text{ than }{M}\qquad \textbf{(E)}\ \text{midway between }{M}\text{ and }{N}$

1953 AMC 12/AHSME, 36

Tags: quadratics

Determine $ m$ so that $ 4x^2\minus{}6x\plus{}m$ is divisible by $ x\minus{}3$. The obtained value, $ m$, is an exact divisor of: $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 64$

2008 Harvard-MIT Mathematics Tournament, 4

Tags: quadratics

Find the real solution(s) to the equation $ (x \plus{} y)^2 \equal{} (x \plus{} 1)(y \minus{} 1)$.

2003 District Olympiad, 3

Tags: algebra , polynomial , quadratics , superior algebra , superior algebra unsolved

Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that: (a) $1+1 \neq 0$; (b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$. [i]Marian Andronache[/i] [Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful. [Edit $2^\circ$] OK, thanks.

1993 Kurschak Competition, 1

Tags: inequalities , quadratics , algebra , number theory unsolved , number theory

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.