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Perfect squares: 2011 USAJMO #1

v_Enhance 225

N Mar 27, 2025 by de-Kirschbaum

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

225 replies

v_Enhance

Apr 28, 2011

de-Kirschbaum

Mar 27, 2025

Bounded Quadratic

worthawholebean 38

N Mar 25, 2025 by SomeonecoolLovesMaths

Source: AIME 2010I Problem 6

Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3$ for all real numbers $x$ , and suppose $P(11) = 181$ . Find $P(16)$ .

38 replies

worthawholebean

Mar 17, 2010

SomeonecoolLovesMaths

Mar 25, 2025

Degree Six Polynomial's Roots

ksun48 42

N Mar 17, 2025 by eg4334

Source: 2014 AIME I Problem 14

Let $m$ be the largest real solution to the equation $\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.$ There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$ . Find $a+b+c$ .

42 replies

ksun48

Mar 14, 2014

eg4334

Mar 17, 2025

Geometric Maximization

CatalystOfNostalgia 20

N Feb 28, 2025 by daijobu

Source: AIME 2008I Problem 14

Let $\overline{AB}$ be a diameter of circle $\omega$ . Extend $\overline{AB}$ through $A$ to $C$ . Point $T$ lies on $\omega$ so that line $CT$ is tangent to $\omega$ . Point $P$ is the foot of the perpendicular from $A$ to line $CT$ . Suppose $AB = 18$ , and let $m$ denote the maximum possible length of segment $BP$ . Find $m^{2}$ .

20 replies

CatalystOfNostalgia

Mar 23, 2008

daijobu

Feb 28, 2025

Inequality with a^2+b^2+c^2+abc=4

cn2_71828182846 70

N Feb 27, 2025 by math90

Source: USAMO 2001 #3

Let $a, b, c \geq 0$ and satisfy $a^2+b^2+c^2 +abc = 4 .$ Show that $0 \le ab + bc + ca - abc \leq 2.$

70 replies

cn2_71828182846

Jun 27, 2004

math90

Feb 27, 2025

Polynomial in Two Variables

E^(pi*i)=-1 21

N Feb 26, 2025 by daijobu

Source: AIME 2008I Problem 13

Let
$p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.$ Suppose that
$\begin{align*}p(0,0) &= p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) \\&= p(1,1) = p(1, - 1) = p(2,2) = 0.\end{align*}$ There is a point $\left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $p\left(\tfrac {a}{c},\tfrac {b}{c}\right) = 0$ for all such polynomials, where $a$ , $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$ . Find $a + b + c$ .

21 replies

E^(pi*i)=-1

Mar 23, 2008

daijobu

Feb 26, 2025

Isosceles Trapezoid

CatalystOfNostalgia 23

N Feb 25, 2025 by daijobu

Source: AIME 2008I Problem 10

Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$ . The diagonals have length $10\sqrt {21}$ , and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$ , respectively. Let $F$ be the foot of the altitude from $C$ to $\overline{AD}$ . The distance $EF$ can be expressed in the form $m\sqrt {n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$ .

23 replies

CatalystOfNostalgia

Mar 23, 2008

daijobu

Feb 25, 2025

Dice rolling

solafidefarms 6

N Feb 11, 2025 by ashays

Source: 2006 AIME II 5

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$ , and $6$ , the probability of obtaining face $F$ is greater than $\frac{1}{6}$ , the probability of obtaining the face opposite is less than $\frac{1}{6}$ , the probability of obtaining any one of the other four faces is $\frac{1}{6}$ , and the sum of the numbers on opposite faces is $7$ . When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$ . Given that the probability of obtaining face $F$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$ .

6 replies

solafidefarms

Mar 28, 2006

ashays

Feb 11, 2025

Cubic, Integer Solutions

BarbieRocks 37

N Feb 5, 2025 by Shreyasharma

For some integer $m$ , the polynomial $x^3-2011x+m$ has the three integer roots $a$ , $b$ , and $c$ . Find $|a|+|b|+|c|$ .

37 replies

BarbieRocks

Mar 18, 2011

Shreyasharma

Feb 5, 2025

Fake Complex Numbers

happiface 29

N Feb 2, 2025 by Shreyasharma

Source: 2014 AIME II Problem #10

Let $z$ be a complex number with $|z| = 2014$ . Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$ . Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$ .

29 replies

happiface

Mar 27, 2014

Shreyasharma

Feb 2, 2025

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