Northeastern Section of the
Mathematical Association of America
2008 Collegiate Mathematics Competition

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Practice Problems for 2007

Date
 Problem
 Solution
Tu, 9/25
 Let
P(x)=(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\cdots +x^{1000}.
  • Find the coefficient of $x^{50}$ in $P(x)$.
  • What is the sum of all the coefficients in this polynomial?
Solution
Th, 9/27
 Find the area of that portion of the $xy$-plane which is enclosed by the curve with equation
|2x-1|+|2x+1|+\frac{4|y|}{\sqrt{3}}=4.
 Solution
M, 10/1
 Let $n$ be any integer greater than 1.  Prove that
1^1\cdot 2^2\cdot 3^3\cdots n^n <\left(\frac{2n+1}{3} \right)^{\frac{n(n+1)}{2}}.
 Solution
W, 10/3
 A list of integers has mode 32 and mean 22. The smallest number on the list is 10. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $(m+10)$, the mean and median of the new list would be 24 and $(m+10)$, respectively. If $m$ were instead replaced by $(m-8)$, the median of the new list would be $m-4$. What is $m$? Solution
F, 10/5
 Prove that if $f$ is continuous, then
\int_0^x f(u)(x-u)\,du = \int_0^x \left(\int_0^u f(t)\,dt \right)\,du

 Solution
W, 10/10
 The number 10 is a base for the positive integers because every positive integer can be written uniquely as
d_n 10^n + d_{n-1}10^{n-1} + \cdots + d_1\cdot 10 + d_0,

where each $d_i$ is one of the digits $0,1,2,\dots 9$. The number $-2$ is a base for all integers using the digits 0 or 1. For example $1101$ represents $-3$ since
1(-2)^3 + 1(-2)^2 + 0(-2) + 1 =-3.

Find the representation in base $-2$ for the decimal number $-2374$. 		Solution
M, 10/15
 Let $p$ be an odd prime. Prove that the integer part of
(\sqrt{5}+2)^p - 2^{p+1}
is divisible by $20p$. 		Solution
W, 10/17
 Suppose you repeatedly toss a fair coin until you get two heads in a row. What is the probability that you stop on the 10th toss? Solution
M, 10/22
 Let $(G,*)$ be a group with the following cancellation property:  $x*a*y=b*a*c$ implies $x*y=b*c$ for all $x$, $y$, $a$, $b$, $c$ in $G$.  Prove that $G$ is abelian (that is, that the operation $*$ is commutative.)
 Solution
Th, 10/25
 Evaluate the integral
\int_0^1\!\int_0^1 e^{\max\{x^2,y^2\}}\,dy\,dx.

 Solution
M, 10/29
 The harmonic mean of a set of positive numbers is the reciprocal of the arithmetic mean (ordinary average) of the reciprocals of the numbers. Find $\lim_{n\to\infty} (H_n/n)$, where $H_n$ is the harmonic mean of the $n$ positive integers $n+1$, $n+2$, $n+3, \dots n+n$.
Solution
W, 10/31
 In a large urn there are 1999 orange balls and 2000 yellow balls.  Next to the urn is a large pile of yellow balls.  The following procedure is performed repeatedly.

Two balls are chosen at random from the urn:
  • If both are yellow, one is put back, the other thrown away;
  • If both are orange, they are both thrown away and a yellow ball from the pile is put into the urn;
  • If they are of different colors, the orange one is put back into the urn and the yellow one is thrown away.
What is the color of the last ball in the urn?
 Solution
F, 11/2
 What is the prime factorization of 1,005,010,010,005,001?
 Solution
M, 11/5
 Rationalize the denominator: $\displaystyle\frac{\sqrt[3]{2}}{\sqrt[3]{2}+\sqrt[3]{3}}$.
 Solution
Th, 11/8
 Evaluate $\displaystyle\lim_{x\to 0} \frac{\sqrt[3]{1+cx}-1}{x}$ where $c$ is a constant.
 Solution
W, 11/14 (Last one!)
 Without expanding it, prove that some digit occurs six times in $x=7^{57}\cdot 11$.
 Solution will appear on Th, 11/15

Practice Problems for 2006

Date
 Problem
 Solution
Th, 10/5
 Find the sum of the series \displaystyle\sum_{n=0}^{\infty} \frac{(x+2)^n}{(n+3)!}. Solution
F, 10/6
 Let $\alpha$, $\beta$, $\gamma$, and $\delta$ be the roots of $x^4+bx^3+cx^2+dx+e=0$. Compute

$(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$

in terms of $b$, $c$, $d$, and $e$. 		Solution
M, 10/9 Let $S$ be a set and let $*$ be a binary operation on $S$ satisfying the laws
 
$x*(x*y)=y$ for all $x$, $y$ in $S$

$(y*x)*x=y$ for all $x$, $y$ in $S$

Show that $*$ is commutative but not necessarily associative. 		Solution
W, 10/11 The probability that the square of a positive integer (in decimal notation) ends with the digit 1 is $2/10$ because out of every 10 numbers, those and only those ending with the digits 1 or 9 have squares ending with 1.  What is the probability that the cube of a positive integer chosen at random ends with the digits 11? Solution
F, 10/13 Find the smallest positive number $A$ such that
\frac{21}{|z^4-5z^2+6|}\le A

for every complex number $z$ on the circle $|z|=3$. Prove that $A$ is the smallest such number. 		Solution
M, 10/16 The horizontal line $y=c$ intersects the curve $y=2x-3x^3$ in the first quadrant as in the figure. Find $c$ so that the areas of the two shaded regions are equal.


Solution
W, 10/18 The sides of a triangle have lengths 4, 5, and 6.  Show that one of its angles is twice another.
 Solution
F, 10/20 If $x>y>0$, show that

$\ln x^2-\ln y^2<\displaystyle\frac{x^2-y^2}{xy}$

 Solution
M, 10/23 Evaluate

$\displaystyle\int_0^{\pi/2} \frac{\sin^{2006}x}{\sin^{2006}x+\cos^{2006}x}\,dx$

Hint: If $f$ is a continuous function, then $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$. 		Solution
W, 10/25 A runner runs around a circular track of radius 100 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 200 m from the center of the track.  How fast is the distance between the friends changing when the distance between them is 200 m?
 Solution
M, 10/30 Does any row of Pascal's triangle contain three consecutive entries that are in the ratio 1:2:3? Recall that the $n$th row ($n=0,1,2,\dots$) consists of the entries

$\displaystyle\left ( \begin{array}{c} n \\ k \end{array}\right )=\frac{n!}{k!(n-k)!} \quad (k=0,1,2,\dots, n)$

 Solution
Th, 11/2 If $x$ and $y$ are positive numbers, show that

$\displaystyle\sqrt{\frac{\left(\frac{x}1 \right)^2+\left(\frac{x+y}{2} \right)^2}{2}} \le \frac{\sqrt{\left(\frac{x}1 \right)^2} + \sqrt{\frac{x^2+y^2}{2}}}{2}$

and find all cases of equality.
Solution
M, 11/6 Find all real functions $f$ such that, for all real $x$,

$f(x+2)=f(x) \quad\text{and}\quad f'(x)=f(x+1)-2$.

 Solution
W, 11/8 Evaluate the limit:

$\displaystyle\lim_{n\to\infty} \left( \frac1{\sqrt{n^2+1^2}} + \frac1{\sqrt{n^2+2^2}} + \cdots + \frac1{\sqrt{n^2+n^2}} \right)$

 Solution
F, 11/10 Evaluate

$\displaystyle\int \frac1{x^7-x}\,dx$.

 Solution
M, 11/13
(last one!)
 If $A$ is a square matrix such that $A^3+4A^2+3A+2I=0$, show that $A$ is invertible. ($I$ denotes the identity matrix and 0 the matrix all of whose entries are zero.)
Solution