2006 SMT/Team Problems

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 $\overline{BC}$.

Solution

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?

Solution

Problem 3

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

Solution

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?

Solution

Problem 5

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

Solution

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 $p(2006)-q(2006)$.

Solution

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 $2,500$?

Solution

Problem 8

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

Solution

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 $\left(\frac{AB}{BC}\right)^2$.

Solution

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.

Solution

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 $\sum_{i=1}^{2006}r_i$.

Solution


(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 $360^\circ$.

Solution

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 $\cos\theta$.

Solution

Problem 14

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

Solution

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

Solution


(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

2006 SMT

2006 SMT/Team