Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2000 SMT/Team Problems

Problem 1

You are given a number, and round it to the nearest thousandth, round this result to the nearest hundredth, and round this result to the nearest tenth. If the final result is .7, what is the smallest number you could have been given? As is customary, 5’s are always rounded up. Give the answer as a decimal.

Solution

Problem 2

The price of a gold ring in a certain universe is proportional to the square of its purity and the cube of its diameter. The purity is inversely proportional to the square of the depth of the gold mine and directly proportional to the square of the price, while the diameter is determined so that it is proportional to the cube root of the price and also directly proportional to the depth of the mine. How does the price vary solely in terms of the depth of the gold mine?

Solution

Problem 3

Find the sum of all integers from 1 to 1000 inclusive which contain at least one 7 in their digits, i.e. find 7 + 17 + ... + 979 + 987 + 997.

Solution

Problem 4

All arrangements of letters VNNWHTAAIE are listed in lexicographic (dictionary) order. If AAEHINNTVW is the first entry, what entry number is VANNAWHITE?

Solution

Problem 5

Given $\cos(\alpha + \beta) + \sin(\alpha-\beta) = 0$, $\tan(\beta) = \frac{1}{2000}$, find $\tan(\alpha)$.

Solution

Problem 6

If $\alpha$ is a root of $x^3-x-1 = 0$, compute the value of $\alpha^{10}+2\alpha^8-\alpha^7-3\alpha^6-3\alpha^5+4\alpha^4+2\alpha^3-4\alpha^4-6\alpha-17$.

Solution

Problem 7

8712 is an integral multiple of its reversal, 2178, as 8712=4*2178. Find another 4-digit number which is a non-trivial integral multiple of its reversal.

Solution

Problem 8

A woman has $$1.58 in pennies, nickels, dimes, quarters, half-dollars and silver dollars. If she has a different number of coins of each denomination, how many coins does she have?

Solution

Problem 9

Find all positive primes of the form $4x^4 + 1$, for $x$ an integer

Solution

Problem 10

How many times per day do at least two of the three hands on a clock coincide?

Solution

Problem 11

Find all polynomials $f(x)$ with integer coefficients such that the coefficients of both $f(x)$ and $[f(x)]^3$ lie in the set $\{0, 1, -1\}$.

Solution

Problem 12

At a dance, Abhinav starts from point $(a, 0)$ and moves along the negative x-direction with speed $v_a$, while Pei-Hsin starts from $(0, b)$ and glides in the negative y-direction with speed $v_b$. What is the distance of closest approach between the two?

Solution

Problem 13

Let $P_1, P_2,...,P_n$ be a convex n-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of n obtained from strictly inside the polygon?

Solution

Problem 14

Define a sequence $<x_n>$ of real numbers by specifying an initial $x_0$ and by the recurrence $x_{n+1} = \frac{1+x_n} {1-x_n}$. Find $x_n$ as a function of $x_0$ and $n$, in closed form. There may be multiple cases.

Solution

Problem 15

$\lim_{n\to\infty} nr \sqrt[2]{1-cos(\frac{2\pi}{n})}$

Solution

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2000_SMT/Team_Problems&oldid=115102"