紫色彗星数学竞赛真题（2003年初中组）

2003 UW-Whitewater Middle  School  Mathematics Meet
Titu Andreescu and Jonathan Kane∗
December 11, 2003
Instructions
Teams may fill in answers to the questions over the next 90 minutes. At any time contestants can click the SUBMIT button to submit their team’s entry. Answers may be submitted multiple times by the same team, but only the last set of answers received before the contest ends will be accepted and graded. Contestants are allowed to work on these problems as a team. No help may be provided by persons not on their team. There is no penalty for guessing. (See official rules for complete contest rules.)
Problem 1
In eight years Henry will be three times the age that Sally was last year.
Twenty five years ago their ages added to 83. How old is Henry now?
Problem 2
What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month?
∗Department of Mathematical and Computer Sciences, University of Wisconsin, Whitewater, Wisconsin 53190
Problem 3
What is the largest integer whose prime factors add to 14?
Problem 4
The lengths of the diagonals of a rhombus are, in inches, two consecutive
integers. The area of the rhombus is 210 sq. in. Find its perimeter, in inches.
Problem 5

Problem 6
Evaluate:

Problem 7
Find the smallest n such that every subset of {1, 2, 3, . . . , 2004} with n elements contains at least two elements that are relatively prime.
Problem 8
Let ABCDEF GHIJKL be a regular dodecagon. Find
Problem 9

Problem 10
How many gallons of a solution which is 15% alcohol do we have to mix with a solution that is 35% alcohol to make 250 gallons of a soluition that is 21% alcohol?

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Problem 11
If

where m and n are positive integrs with no common divisor, find m + n.
Problem 12
How many triangles appear in the diagram below?

Problem 13
Let P(x) be a polynomial such that, when divided by x − 2, the remainder is 3 and, when divided by x − 3, the remainder is 2. If, when divided by (x − 2)(x − 3), the remainder is ax + b, find a² + b².

Problem 14
Let a, b, c be real numbers such that a² − 2 = 3b − c, b² + 4 = 3 + a, and c² + 4 = 3a − b. Find a⁴ + b⁴ + c⁴.
Problem 15

Problem 16
Find the largest real number x such that

Problem 17

Problem 18
A circle radius 320 is tangent to the inside of a circle radius 1000. The smaller circle is tangent to a diameter of the larger circle at a point P. How far is the point P from the outside of the larger circle?
Problem 19
Let x₁ and x₂ be the roots of the equation x² + 3x + 1 = 0. Compute

Problem 20
In how many ways can we form three teams of four players each from a group
of 12 participants?
Problem 21

Problem 22

Problem 23
For each positive integer m and n define function f(m, n) by f(1, 1) = 1,f(m + 1, n) = f(m, n) + m and f(m, n + 1) = f(m, n) all the values of p such that f(p, q) = 2004 for some q.− n. Find the sum of all the values of p such that f(p, q) = 2004 for some q.
Problem 24

Problem 25
Given that (1 + tan 1^◦)(1 + tan 2^◦). . .(1 + tan 45^◦) = 2ⁿ, find n.