Olympiad problems

1969 Canada National Olympiad, 7

Tags: abstract algebra , polynomial , quadratic , domain , modular arithmetic , algebra , function

Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.

2006 USAMO, 4

Tags: quadratic , algebra

Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$

2015 Saint Petersburg Mathematical Olympiad, 4

Tags: quadratic , number theory

A positive integer $n$ is called [i]Olympic[/i], if there exists a quadratic trinomial with integer coeffecients $f(x)$ satisfying $f(f(\sqrt{n}))=0$. Determine, with proof, the largest Olympic number not exceeding $2015$. [i]A. Khrabrov[/i]

2004 Iran Team Selection Test, 1

Tags: quadratic , special factorizations , modular arithmetic , number theory

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2012 AMC 12/AHSME, 10

Tags: ellipse , geometry , quadratic , conic , analytic geometry

What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$? ${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $

2007 Iran MO (3rd Round), 6

Tags: quadratic , complex number , polynomial , algebra

Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$. [img]http://i14.tinypic.com/4tk6mnr.png[/img] a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.

1969 IMO Shortlist, 63

Tags: number theory , quadratic , additive number theory

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

2009 India National Olympiad, 6

Tags: algebra , quadratic , inequality , polynomial

Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that: $ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.

1977 AMC 12/AHSME, 21

Tags: quadratic , special factorizations

For how many values of the coeﬃcient $a$ do the equations \begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*} have a common real solution? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

2013 Hitotsubashi University Entrance Examination, 4

Tags: quadratic , geometry

Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of $t$ such that the maximum value of $OP$ is 3.

2013 Online Math Open Problems, 28

Tags: quadratic , modular arithmetic

Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that (i) $\frac{p+1}{2}$ is even but is not a power of $2$, and (ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$. [i]Proposed by Evan Chen[/i]

2015 AMC 10, 12

Tags: quadratic formula , quadratic , vieta , absolute value , algebra , polynomial

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

2003 IberoAmerican, 3

Tags: number theory , quadratic , induction , modular arithmetic

The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\] Show that $2003$ is not divisor of any of the terms in these two sequences.

1988 IMO Longlists, 36

Tags: quadratic , algebra , geometry , ratio

[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio \[ \frac{EG}{GF} \] [b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$

1983 AIME Problems, 3

Tags: domain , algebra , vieta , quadratic , polynomial , function

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

2008 Serbia National Math Olympiad, 1

Tags: quadratic , number theory

Find all nonegative integers $ x,y,z$ such that $ 12^x\plus{}y^4\equal{}2008^z$

2024 All-Russian Olympiad Regional Round, 9.2

Tags: quadratic , function graphing , conic , parabola , trapezoid , analytic geometry , geometry

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

2004 USA Team Selection Test, 1

Tags: polynomial , quadratic , algebra , inequalities , parallelogram , trigonometry , geometry

Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that \[ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. \] Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.

2011 Today's Calculation Of Integral, 694

Tags: quadratic , logarithm , integration , inequalities , calculus computations , calculus

Prove the following inequality: \[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\] created by kunny

2016 SDMO (High School), 1

Tags: vieta , quadratic , 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.

2014 Iran Team Selection Test, 4

Tags: number theory , quadratic

$n$ 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.