Art of Problem Solving

AoPS Online

2006 SMT/Team Problems

Contents

1 Problem 1
2 Problem 2
3 Problem 3
4 Problem 4
5 Problem 5
6 Problem 6
7 Problem 7
8 Problem 8
9 Problem 9
10 Problem 10
11 Problem 11
12 Problem 12
13 Problem 13
14 Problem 14
15 Problem 15
16 See Also

Problem 1

Given $\triangle ABC$ , where $A$ is at $(0,0)$ , $B$ is at $(20,0)$ , and $C$ is on the positive $y$ axis. Cone $M$ is formed when $\triangle ABC$ is rotated about the $x$ axis, and cone $N$ is formed when $\triangle ABC$ is rotated about the $y$ axis. If the volume of cone $M$ minus the volume fo cone $N$ is $140\pi$ , find the length of $\overline{BC}$ .

Problem 2

In a given sequence $\{S_1, S_2, \cdots S_k\}$ , for terms $n\ge3$ , $S_n=\sum_{i=1}^{n-1}i\cdot S_{n-i}$ . For example, if the first two elements are $2$ and $3$ , respectively, the third entry would be $1\cdot3+2\cdot2=7$ , and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$ , and so on. Given that a sequence of integers having this form starts with $2$ , and the $7\text{th}$ element is $68$ , what is the second element?

Problem 3

A triangle has altitudes of lengths $5$ and $7$ . What is the maximum length of the third altitude?

Problem 4

Let $x+y=a$ and $xy=b$ . The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$ . What is this polynomial?

Problem 5

There exist two positive numbers $x$ such that $\sin(\arccos(\tan(\arcsin(x))))=x$ . Find the product of the two possible $x$ .

Problem 6

The expression $16^n+4^n+1$ is equivalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\frac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$ .

Problem 7

Let $S$ be the set of all $3$ -tuples $(a, b, c)$ that satisfy $a+b+c=3000$ and $a,b,c>0$ . If one of these $3$ -tuples is chosen at random, what's the probability that $a, b$ or $c$ is greater than or equal to $2,500$ ?

Problem 8

Evaluate: $\lim_{n\to\infty}\sum_{k=n^2}^{(n+1)^2}\frac{1}{\sqrt{k}}$

Problem 9

$\triangle ABC$ has $AB=AC$ . Points $M$ and $N$ are midpoints of $\overline{AB}$ and $\overline{AC}$ , respectively. The medians $\overline{MC}$ and $\overline{NB}$ intersect at a right angle. Find $\left(\frac{AB}{BC}\right)^2$ .

Problem 10

Find the smallest positive integer $m$ for which there are at least $11$ even and $11$ odd positive integers $n$ so that $\frac{n^3+m}{n+2}$ is an integer.

Problem 11

Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\cdots+c_1x+c_0$ has roots $r_1, r_2, \cdots, r_{2006}$ . The coefficients satisfy $2i\frac{c_i}{c_{2006}-i}=2j\frac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i, j\le2006$ . Given that $\sum_{i\not=j, i=1, j=1}^{2006}\frac{r_i}{r_j}=42$ , determine $\sum_{i=1}^{2006}r_i$ .

(Note: The original problem asked for $\sum_{i=1}^{2006}(r_1+r_2+\cdots+r_{2006})$ , but the official solution makes it clear that the actual desired sum is $\sum_{i=1}^{2006}r_i$ .)

Problem 12

Find the total number of $k$ -tuples $(n_1, n_2, \cdots, n_k)$ of positive integers so that $n_{i+1}\ge n_i$ for each $i$ , and $k$ regular polygons with numbers of sides $n_1, n_2, \cdots n_k$ respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is $360^\circ$ .

Problem 13

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$ axis. Compute $\cos\theta$ .

Problem 14

Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$ .

Problem 15

Let $c_i$ denote the $i\text{th}$ composite integer so that $\{c_i\}=4, 6, 8, 9\cdots$ . Compute

$\prod_{i=1}^{\infty}\frac{c_i^2}{c_i^2-1}$

(Hint: $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ )

(Note: The original hint stated that $\sum_{i=1}^{n}\frac{1}{n^2}=\frac{\pi^2}{6}$ .)

See Also

Stanford Mathematics Tournament

SMT Problems and Solutions

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_SMT/Team_Problems&oldid=47185"

© 2024 AoPS Incorporated