Olympiad problems

2002 India IMO Training Camp, 3

Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?

2012 National Olympiad First Round, 19

Tags: quadratics , algebra

What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

1952 AMC 12/AHSME, 40

Tags: quadratics , function

In order to draw a graph of $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $ x$ were $ 3844$, $ 3969$, $ 4096$, $ 4227$, $ 4356$, $ 4489$, $ 4624$, and $ 4761$. The one which is incorrect is: $ \textbf{(A)}\ 4096 \qquad\textbf{(B)}\ 4356 \qquad\textbf{(C)}\ 4489 \qquad\textbf{(D)}\ 4761 \qquad\textbf{(E)}\ \text{none of these}$

PEN A Problems, 8

Tags: quadratics , search , modular arithmetic , algebra , Divisibility Theory , pen

The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

2008 Harvard-MIT Mathematics Tournament, 6

Tags: quadratics , inequalities , function , conics , parabola , analytic geometry , absolute value

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2013 Romanian Master of Mathematics, 1

Tags: floor function , modular arithmetic , quadratics , number theory

For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.

2008 Bulgaria National Olympiad, 1

Tags: quadratics , number theory unsolved , number theory

Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number.

2012 AMC 12/AHSME, 16

Tags: ARML , AMC , AMC 12 , trigonometry , quadratics , geometry , circumcircle

Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqrt{7}\qquad\textbf{(E)}\ \sqrt{30} $

2011 Morocco National Olympiad, 2

Tags: quadratics , algebra unsolved , algebra

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

1957 AMC 12/AHSME, 45

Tags: inequalities , Gauss , quadratics

If two real numbers $ x$ and $ y$ satisfy the equation $ \frac{x}{y} \equal{} x \minus{} y$, then: $ \textbf{(A)}\ {x \ge 4}\text{ and }{x \le 0}\qquad \\ \textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\ \textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad \\ \textbf{(D)}\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad \\ \textbf{(E)}\ \text{both }{x}\text{ and }{y}\text{ must be rational}$

1995 All-Russian Olympiad, 3

Tags: quadratics , algebra , polynomial , symmetry , function , complex numbers , algebra proposed

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

1994 Romania TST for IMO, 1:

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

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

1971 Canada National Olympiad, 2

Tags: quadratics

Let $x$ and $y$ be positive real numbers such that $x+y=1$. Show that \[ \left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9. \]

2011 Today's Calculation Of Integral, 694

Tags: calculus , integration , inequalities , logarithms , quadratics , calculus computations

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

2006 Pre-Preparation Course Examination, 2

Tags: algebra , polynomial , trigonometry , quadratics , algebra proposed

a) Show that you can divide an angle $\theta$ to three equal parts using compass and ruler if and only if the polynomial $4t^3-3t-\cos (\theta)$ is reducible over $\mathbb{Q}(\cos (\theta))$. b) Is it always possible to divide an angle into five equal parts?

1990 India National Olympiad, 2

Tags: calculus , integration , quadratics , number theory proposed , number theory , Diophantine equation

Determine all non-negative integral pairs $ (x, y)$ for which \[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]

2010 All-Russian Olympiad, 1

Tags: induction , topology , geometry , 3D geometry , sphere , quadratics , algebra solved

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

2007 All-Russian Olympiad, 1

Tags: quadratics , algebra unsolved , algebra

Unitary quadratic trinomials $ f(x)$ and $ g(x)$ satisfy the following interesting condition: $ f(g(x)) \equal{} 0$ and $ g(f(x)) \equal{} 0$ do not have real roots. Prove that at least one of equations $ f(f(x)) \equal{} 0$ and $ g(g(x)) \equal{} 0$ does not have real roots too. [i]S. Berlov [/i]

PEN C Problems, 4

Tags: floor function , quadratics , inequalities , number theory , prime numbers , Quadratic Residues

Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

1979 Bulgaria National Olympiad, Problem 4

Tags: quadratics , parameterization , function , Polynomials , algebra

For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation $$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.

2007 Putnam, 4

Tags: Putnam , algebra , polynomial , logarithms , quadratics , college contests

A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$

2008 ISI B.Math Entrance Exam, 8

Tags: quadratics , trigonometry , algebra proposed , algebra

Let $a^2+b^2=1$ , $c^2+d^2=1$ , $ac+bd=0$ Prove that $a^2+c^2=1$ , $b^2+d^2=1$ , $ab+cd=0$ .

2010 Princeton University Math Competition, 4

Tags: quadratics , calculus , integration , algebra , quadratic formula

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

2013 Iran Team Selection Test, 5

Tags: modular arithmetic , algebra , polynomial , quadratics , Vieta , number theory proposed , number theory

Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

2022 AIME Problems, 1

Tags: AMC , AIME , AIME I , quadratics

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 ${P(0) + Q(0)}$.