Olympiad problems

2005 AIME Problems, 7

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.

1991 Hungary-Israel Binational, 4

Tags: algebra , system of equations , quadratic

Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

2018 CCA Math Bonanza, T10

Tags: quadratic

The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$. [i]2018 CCA Math Bonanza Team Round #10[/i]

2020 CCA Math Bonanza, T5

Tags: quadratic

Find all pairs of real numbers $(x,y)$ satisfying both equations \[ 3x^2+3xy+2y^2 =2 \] \[ x^2+2xy+2y^2 =1. \] [i]2020 CCA Math Bonanza Team Round #5[/i]

PEN S Problems, 4

Tags: binomial coefficient , algebra , quadratic , induction

If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.

1997 All-Russian Olympiad, 1

Tags: modular arithmetic , algebra , quadratic , polynomial

Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

2007 Junior Balkan Team Selection Tests - Moldova, 6

Tags: angle , quadratic , trinomial

The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$

2013 JBMO TST - Turkey, 4

Tags: polynomial , algebra , inequalities , quadratic

For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that \[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]

2010 IMC, 4

Tags: number theory , function , abstract algebra , quadratic , set theory

Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.

2010 South East Mathematical Olympiad, 1

Tags: algebra , quadratic , number theory

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

2013 Today's Calculation Of Integral, 867

Tags: calculus , derivative , integration , function , quadratic , linear equation , algebra

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2014-2015 SDML (High School), 15

Tags: floor function , quadratic

Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$. $\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$

1979 Romania Team Selection Tests, 4.

Tags: polynomial , inequalities , algebra , quadratic

Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that \[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \; \left|P(x)+\frac{1}{x-4}\right| \leqslant 0.01.\] Are there linear polynomials with this property? [i]Octavian Stănășilă[/i]

1995 Flanders Math Olympiad, 2

Tags: function , calculus , algebra , quadratic , integration

How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?

1999 AIME Problems, 3

Tags: polynomial , algebra , vieta , quadratic , special factorizations

Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.

2014 IFYM, Sozopol, 2

Tags: number theory , quadratic

Polly can do the following operations on a quadratic trinomial: 1) Swapping the places of its leading coefficient and constant coefficient (swapping $a_2$ with $a_0$); 2) Substituting (changing) $x$ with $x-m$, where $m$ is an arbitrary real number; Is it possible for Polly to get $25x^2+5x+2014$ from $6x^2+2x+1996$ with finite applications of the upper operations?

2010 AIME Problems, 9

Tags: inequalities , function , 3d geometry , quadratic

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

2009 AMC 8, 23

Tags: algebra , quadratic , quadratic formula

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $ 400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? $ \textbf{(A)}\ 26 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 34$

2011 Tuymaada Olympiad, 4

Tags: polynomial , algebra , quadratic , analytic number theory , integer sequence , integer polynomial

Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.

2015 India National Olympiad, 5

Tags: parallelogram , analytic geometry , trigonometry , geometry , vector , quadratic

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

2022 AIME Problems, 1

Tags: quadratic

Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients of $2$ and $-2$, respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53)$. Find ${P(0) + Q(0)}$.

2011 ISI B.Math Entrance Exam, 2

Tags: number theory , 3d geometry , quadratic

Given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that $r=s$.

1965 AMC 12/AHSME, 34

Tags: function , quadratic , calculus

For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$

2013 APMO, 2

Tags: floor function , algebra , function , modular arithmetic , quadratic , inequalities

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2010 Dutch BxMO TST, 5

Tags: number theory , quadratic , permutation , perfect square

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.