Olympiad problems

2008 Harvard-MIT Mathematics Tournament, 1

Tags: quadratics

Determine all pairs $ (a,b)$ of real numbers such that $ 10, a, b, ab$ is an arithmetic progression.

2007 Purple Comet Problems, 2

Tags: ratio , quadratics , number theory , relatively prime , algebra , quadratic formula

A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.

2010 Math Prize For Girls Problems, 16

Tags: quadratics , function , analytic geometry , algebra , polynomial

Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies \[ \sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3}, \] compute the ordered triple $(a, b, c)$.

2007 AMC 10, 20

Tags: quadratics , AMC

Suppose that the number $ a$ satisfies the equation $ 4 \equal{} a \plus{} a^{ \minus{} 1}$. What is the value of $ a^{4} \plus{} a^{ \minus{} 4}$? $ \textbf{(A)}\ 164 \qquad \textbf{(B)}\ 172 \qquad \textbf{(C)}\ 192 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 212$

PEN O Problems, 46

Tags: modular arithmetic , quadratics

Suppose $p$ is a prime with $p \equiv 3 \; \pmod{4}$. Show that for any set of $p-1$ consecutive integers, the set cannot be divided two subsets so that the product of the members of the one set is equal to the product of the members of the other set.

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: quadratics , algebra , equations

Find the real numbers $ m $ which have the property that the equation $$ x^2-2mx+2m^2=25 $$ has two integer solutions.

2011 Mexico National Olympiad, 3

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

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system: \[a_1^2 + a_1 - 1 = a_2\] \[ a_2^2 + a_2 - 1 = a_3\] \[\hspace*{3.3em} \vdots \] \[a_{n}^2 + a_n - 1 = a_1\]

2014 Lithuania Team Selection Test, 3

Tags: linear algebra , matrix , quadratics , algebra , system of equations , algebra solved

Given such positive real numbers $a, b$ and $c$, that the system of equations: $ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $ has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.

2006 Romania Team Selection Test, 1

Tags: induction , quadratics , algebra , algebra proposed

Let $\{a_n\}_{n\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, \[ a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . \] a) Prove that all the terms of the sequence are positive integers. b) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$. [i]Valentin Vornicu[/i]

1988 China Team Selection Test, 1

Tags: inequalities , quadratics , inequalities unsolved

Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds: \[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\] Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).

1969 AMC 12/AHSME, 14

Tags: quadratics , inequalities , AMC

The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that: $\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\, \textbf{(B) }x>2\text{ or }x<-2$ $\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\, \textbf{(D) }x>1\text{ or }x<-2\qquad$ $\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$

2004 Croatia Team Selection Test, 1

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

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

2014 Saudi Arabia BMO TST, 1

Tags: quadratics , algebra unsolved , algebra

Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers

2020 Brazil Undergrad MO, Problem 6

Tags: function , college contests , quadratics , roots , Fixed point , Brazilian Undergrad MO , Brazilian Undergrad MO 2020

Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

2011 AMC 12/AHSME, 24

Tags: geometry , perimeter , quadratics , algebra , polynomial , quadratic formula , AMC

Let $P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3}+7)$. What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$? $ \textbf{(A)}\ 4\sqrt{3}+4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2}+3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2}+4\sqrt{3} \qquad $ $\textbf{(E)}\ 4\sqrt{3}+6 $

2009 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: quadratics , algebra unsolved , algebra

Find all pairs $(a, b)$ of real numbers with the following property: [list]Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.[/list]

2006 Turkey Team Selection Test, 1

Tags: geometry , circumcircle , quadratics , trigonometry , Pythagorean Theorem , algebra , geometry proposed

Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.

1991 IberoAmerican, 6

Tags: geometry , circumcircle , quadratics , parallelogram , algebra , geometry proposed

Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.

2008 National Olympiad First Round, 19

Tags: function , quadratics , algebra , functional equation

Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that \[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \] for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{There is more than one solution} $

2004 Indonesia MO, 2

Tags: quadratics , Vieta , algebra

Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.

2013 Indonesia MO, 6

Tags: quadratics , number theory unsolved , number theory

A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$. a. Prove that $2013$ is strong. b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.

1984 Vietnam National Olympiad, 1

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

$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$. $(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.

2014 Online Math Open Problems, 23

Tags: Online Math Open , modular arithmetic , number theory , greatest common divisor , quadratics , Law of Cosines

For a prime $q$, let $\Phi_q(x)=x^{q-1}+x^{q-2}+\cdots+x+1$. Find the sum of all primes $p$ such that $3 \le p \le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying \[\dbinom{N}{\Phi_q(p)}\equiv \dbinom{2\Phi_q(p)}{N} \not \equiv 0 \pmod p. \][i]Proposed by Sammy Luo[/i]

2010 Purple Comet Problems, 23

Tags: symmetry , quadratics , number theory , relatively prime , Pythagorean Theorem , geometry , algebra

A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4)); [/asy]

2010 Stanford Mathematics Tournament, 19

Tags: algebra , polynomial , quadratics , quadratic formula

Find the roots of $6x^4+17x^3+7x^2-8x-4$