Olympiad problems

2005 Romania Team Selection Test, 3

Tags: modular arithmetic , floor function , quadratics , induction , algebra , functional equation , number theory proposed

Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that \[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \] [i]Călin Popescu[/i]

2019 Belarus Team Selection Test, 2.1

Tags: quadratics , algebra , polynomial

Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$. Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that $$ p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1. $$ [i](I. Gorodnin)[/i]

2003 Korea Junior Math Olympiad, 2

Tags: Integer , quadratics equation , algebra , number theory , quadratics

$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation $$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.

PEN H Problems, 54

Tags: calculus , integration , ratio , geometry , perimeter , quadratics , LaTeX

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.

2000 AIME Problems, 8

Tags: geometry , trapezoid , ratio , quadratics , trigonometry , algebra , quadratic formula

In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$

2013 Finnish National High School Mathematics Competition, 1

Tags: algebra , polynomial , quadratics , algebra unsolved

The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.

2012 Stanford Mathematics Tournament, 8

Tags: quadratics

For real numbers $(x, y, z)$ satisfying the following equations, ﬁnd all possible values of $x+y+z$ $x^2y+y^2z+z^2x=-1$ $xy^2+yz^2+zx^2=5$ $xyz=-2$

2006 ISI B.Stat Entrance Exam, 2

Tags: quadratics

Suppose that $a$ is an irrational number. (a) If there is a real number $b$ such that both $(a+b)$ and $ab$ are rational numbers, show that $a$ is a quadratic surd. ($a$ is a quadratic surd if it is of the form $r+\sqrt{s}$ or $r-\sqrt{s}$ for some rationals $r$ and $s$, where $s$ is not the square of a rational number). (b) Show that there are two real numbers $b_1$ and $b_2$ such that i) $a+b_1$ is rational but $ab_1$ is irrational. ii) $a+b_2$ is irrational but $ab_2$ is rational. (Hint: Consider the two cases, where $a$ is a quadratic surd and $a$ is not a quadratic surd, separately).

2021 German National Olympiad, 1

Tags: quadratics , algebra , algebra proposed , quadratic equation , roots

Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: floor function , ratio , function , quadratics , calculus , integration , geometric series

Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.

PEN I Problems, 11

Tags: floor function , quadratics , function , pen

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

2012 Belarus Team Selection Test, 2

Tags: geometry , circumcircle , algebra , polynomial , quadratics , complex numbers , IMO Shortlist

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

1958 AMC 12/AHSME, 39

Tags: quadratics , function , absolute value

We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that: $ \textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad \textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$

2007 All-Russian Olympiad, 6

Tags: algebra , polynomial , calculus , integration , quadratics , algebra proposed

Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots? [i]N. Agakhanov, I. Bogdanov[/i]

2012 Vietnam National Olympiad, 2

Tags: quadratics , algebra , polynomial , arithmetic sequence , algebra proposed

Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.

1986 China Team Selection Test, 3

Tags: inequalities , quadratics , inequalities unsolved

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

1969 IMO Shortlist, 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.

2006 Team Selection Test For CSMO, 1

Tags: LaTeX , modular arithmetic , quadratics , number theory , relatively prime , algebra , difference of squares

Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

2007 District Olympiad, 4

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

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]

2003 Purple Comet Problems, 16

Tags: quadratics , algebra , quadratic formula

Find the largest real number $x$ such that \[\left(\dfrac{x}{x-1}\right)^2+\left(\dfrac{x}{x+1}\right)^2=\dfrac{325}{144}.\]

2002 Baltic Way, 16

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

Find all nonnegative integers $m$ such that \[a_m=(2^{2m+1})^2+1 \] is divisible by at most two different primes.

PEN D Problems, 11

Tags: quadratics , induction , modular arithmetic , symmetry , Congruences

During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.

2007 Romania Team Selection Test, 3

Tags: inequalities , quadratics , combinatorics proposed , combinatorics

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

2009 Princeton University Math Competition, 1

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

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.

2002 Iran MO (3rd Round), 16

Tags: quadratics , Iran , Inequality , algebra

For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$