Olympiad problems

2013 Princeton University Math Competition, 8

Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$.

2011 USAJMO, 1

Tags: modular arithmetic , quadratic , number theory

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

2006 National Olympiad First Round, 30

Tags: quadratic , modular arithmetic

How many integer triples $(x,y,z)$ are there such that \[\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}\] where $0\leq x < 13$, $0\leq y <13$, and $0\leq z< 13$? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 23 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 49 \qquad\textbf{(E)}\ \text{None of above} $

1969 AMC 12/AHSME, 34

Tags: quadratic , polynomial , algebra

The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as: $\textbf{(A) }2^{100}-1\qquad \textbf{(B) }2^{100}(x-1)-(x-2)\qquad \textbf{(C) }2^{100}(x-3)\qquad$ $\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad \textbf{(E) }2^{100}(x+1)-(x+2)$

2023 Auckland Mathematical Olympiad, 5

Tags: algebra , quadratic , combinatorics

There are $11$ quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this $$\star x^2 + \star x + \star= 0.$$ Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number. The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible. What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players.

2002 AMC 12/AHSME, 23

Tags: algebra , quadratic , complex number , quadratic formula , rational root theorem , polynomial

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$. $\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$

2003 District Olympiad, 2

Tags: algebra , quadratic , quadratic formula

Find $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$, and the digits $\displaystyle a_1,a_2,\ldots,a_n$ such that \[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]

1957 AMC 12/AHSME, 2

Tags: quadratic

In the equation $ 2x^2 \minus{} hx \plus{} 2k \equal{} 0$, the sum of the roots is $ 4$ and the product of the roots is $ \minus{}3$. Then $ h$ and $ k$ have the values, respectively: $ \textbf{(A)}\ 8\text{ and }{\minus{}6} \qquad \textbf{(B)}\ 4\text{ and }{\minus{}3}\qquad \textbf{(C)}\ {\minus{}3}\text{ and }4\qquad \textbf{(D)}\ {\minus{}3}\text{ and }8\qquad \textbf{(E)}\ 8\text{ and }{\minus{}3}$

2006 China Second Round Olympiad, 7

Tags: quadratic , trigonometry

Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$.

1991 India National Olympiad, 7

Tags: quadratic , complex number , rational root theorem , system of equations , polynomial , algebra

Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

2022 Girls in Math at Yale, 9

Tags: quadratic , college

Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$. [i]Proposed by Andrew Wu[/i] (Note: wording changed from original to specify that $a \neq 20, 22$.)

2007 Tournament Of Towns, 4

Tags: quadratic , polynomial , algebra

Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?

2012 Turkey Junior National Olympiad, 1

Tags: number theory , integration , quadratic , diophantine equation , calculus

Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

2013 AIME Problems, 14

Tags: trig identities , number theory , relatively prime , geometric series , quadratic , ratio , trigonometry

For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2005 Germany Team Selection Test, 1

Tags: quadratic , sequence , relatively prime , number theory

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

2002 Iran Team Selection Test, 12

Tags: quadratic , combinatorics , group theory , induction

We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic. [i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.

1969 IMO Longlists, 44

Tags: radius , quadratic , circumcircle , geometry

$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.

2010 India IMO Training Camp, 2

Tags: quadratic , polynomial , algebra

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

2001 Slovenia National Olympiad, Problem 2

Tags: quadratic

Find all rational numbers $r$ such that the equation $rx^2 + (r + 1)x + r = 1$ has integer solutions.

2005 AMC 12/AHSME, 12

Tags: algebra , quadratic

The quadratic equation $ x^2 \plus{} mx \plus{} n \equal{} 0$ has roots that are twice those of $ x^2 \plus{} px \plus{} m \equal{} 0$, and none of $ m,n,$ and $ p$ is zero. What is the value of $ n/p$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

2015 Saint Petersburg Mathematical Olympiad, 1

Tags: quadratic , algebra , polynomial

Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ? [i]A. Khrabrov[/i]

1990 All Soviet Union Mathematical Olympiad, 516

Tags: rational , algebra , quadratic , root , trinomial

Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.

2001 Pan African, 3

Tags: quadratic , perpendicular bisector , algebra , geometry

Let $S_1$ be a semicircle with centre $O$ and diameter $AB$.A circle $C_1$ with centre $P$ is drawn, tangent to $S_1$, and tangent to $AB$ at $O$. A semicircle $S_2$ is drawn, with centre $Q$ on $AB$, tangent to $S_1$ and to $C_1$. A circle $C_2$ with centre $R$ is drawn, internally tangent to $S_1$ and externally tangent to $S_2$ and $C_1$. Prove that $OPRQ$ is a rectangle.

2018 Canadian Mathematical Olympiad Qualification, 3

Tags: geometry , quadratic

Let $ABC$ be a triangle with $AB = BC$. Prove that $\triangle ABC$ is an obtuse triangle if and only if the equation $$Ax^2 + Bx + C = 0$$ has two distinct real roots, where $A$, $B$, $C$, are the angles in radians.

1979 Bulgaria National Olympiad, Problem 4

Tags: function , polynomial , quadratic , parameterization , 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.