Olympiad problems

2009 Princeton University Math Competition, 1

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

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 $a + b + c$.

1991 India Regional Mathematical Olympiad, 6

Tags: quadratic , quadratic formula , algebra

Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers.

2014 India IMO Training Camp, 1

Tags: quadratic formula , quadratic , algebra

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

2011 Purple Comet Problems, 9

Tags: polynomial , quadratic , quadratic formula , sum of roots , product of roots , algebra

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

2006 Cezar Ivănescu, 1

Tags: quadratic formula , cristinel mortici , monotone , equation , algebra

Solve the equation [b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $ [b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $ [i]Cristinel Mortici[/i]

1992 AMC 12/AHSME, 28

Tags: quadratic formula , quadratic , complex number , algebra

Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is $ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $

2005 AIME Problems, 4

Tags: algebra , quadratic formula , special factorizations , geometry , rectangle , quadratic , difference of squares

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.

2011 AMC 12/AHSME, 24

Tags: geometry , perimeter , quadratic , quadratic formula , polynomial , algebra

Let $P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3}+7)$. What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$? $ \textbf{(A)}\ 4\sqrt{3}+4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2}+3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2}+4\sqrt{3} \qquad $ $\textbf{(E)}\ 4\sqrt{3}+6 $

1993 AMC 12/AHSME, 20

Tags: algebra , quadratic , complex number , quadratic formula

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true? $ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $

2006 Greece Junior Math Olympiad, 4

Tags: quadratic formula , function , domain , quadratic , algebra

If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$ [u]Babis[/u] [b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]

2010 Stanford Mathematics Tournament, 4

Tags: quadratic formula , quadratic , algebra , ratio

Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: algebra , arithmetic sequence , quadratic , arithmetic series , quadratic formula , polynomial , function

Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$. A. 1 B. 3 C. 7 D. 12 E. None of these

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 \[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]

2010 AMC 12/AHSME, 15

Tags: probability , quadratic , quadratic formula , algebra

A coin is altered so that the probability that it lands on heads is less than $ \frac {1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $ \frac {1}{6}$. What is the probability that the coin lands on heads? $ \textbf{(A)}\ \frac {\sqrt {15} \minus{} 3}{6}\qquad \textbf{(B)}\ \frac {6 \minus{} \sqrt {6\sqrt {6} \plus{} 2}}{12}\qquad \textbf{(C)}\ \frac {\sqrt {2} \minus{} 1}{2}\qquad \textbf{(D)}\ \frac {3 \minus{} \sqrt {3}}{6}\qquad \textbf{(E)}\ \frac {\sqrt {3} \minus{} 1}{2}$

2001 All-Russian Olympiad, 1

Tags: polynomial , quadratic , quadratic formula , algebra

The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.

2000 Junior Balkan MO, 1

Tags: quadratic , quadratic formula , algebra

Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$.

2010 Stanford Mathematics Tournament, 19

Tags: quadratic , quadratic formula , algebra , polynomial

Find the roots of $6x^4+17x^3+7x^2-8x-4$

Oliforum Contest IV 2013, 2

Tags: geometry , power of a point , circumcircle , quadratic , quadratic formula , algebra

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

2000 AIME Problems, 8

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

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 $BC^2.$

2019 AMC 12/AHSME, 12

Tags: logarithm , quadratic formula , algebra

Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$? $\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$

1971 AMC 12/AHSME, 19

Tags: algebra , conic , ellipse , quadratic , quadratic formula

If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is $\textbf{(A) }\textstyle\frac{1}{2}\qquad\textbf{(B) }\frac{2}{3}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$

1960 AMC 12/AHSME, 39

Tags: quadratic , quadratic formula , algebra

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be: $ \textbf{(A)}\ \text{both rational} \qquad\textbf{(B)}\ \text{both real but not rational} \qquad\textbf{(C)}\ \text{both not real}\qquad$ $\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real} $

2013 AMC 12/AHSME, 6

Tags: quadratic , special factorizations , quadratic formula , algebra

Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $

2009 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: number theory , quadratic , quadratic formula , algebra

Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.