紫色彗星数学竞赛真题（2004年初中组）

Spring 2004 UW-Whitewater Middle School
Mathematics Meet
Titu Andreescu and Jonathan Kane∗
March 18, 2004

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
This year February 29 fell on a Sunday. In what year will February 29 next fall on a Sunday?
Problem 2

∗Department of Mathematical and Computer Sciences, University of Wisconsin, Whitewater, Wisconsin 53190
Problem 3
In triangle ABC, three lines are drawn parallel to side BC dividing the altitude of the triangle into four equal parts. If the area of the second largest part is 35, what is the area of the whole triangle ABC ?

Problem 4

Problem 5
Write the number 2004₍₅₎ [2004 base 5] as a number in base 6.
Problem 6
How many different positive integers divide 10! ?
Problem 7
A rectangle has area 1100. If the length is increased by ten percent and the width is decreased by ten percent, what is the area of the new rectangle?
Problem 8
The number 2.5081081081081... can be written as m/n where m and n are natural numbers with no common factors. Find m + n.
Problem 9
How many positive integers less that 200 are relatively prime to either 15 or 24?
Problem 10
One rainy afternoon you write the number 1 once, the number 2 twice, the number 3 three times, and so forth until you have written the number 99 ninety-nine times. What is the 2005th digit that you write?

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Problem 11
Find the sum of all integers x satisfying 1 + 8x ≤ 358 − 2x ≤ 6x + 94.
Problem 12
If f(x, y) = xy + 2x + y + 1, find f(f(2, f(3, 4)), 5).
Problem 13
How many three digit numbers are made up of three distinct digits?
Problem 14
A polygon has five times as many diagonals as it has sides. How many vertices does the polygon have?
Problem 15
Find the prime number p for which p + 2500 is a perfect square.
Problem 16
A week ago, Sandy’s seasonal Little League batting average was 360. After five more at bats this week, Sandy’s batting average is up to 400. What is the smallest number of hits that Sandy could have had this season?
Problem 17

Problem 18
Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples:

Problem 19
Find n such that n − 76 and n + 76 are both cubes of positive integers.
Problem 20

Problem 21
Find the number of different quadruples (a, b, c, d) of positive integers such that ab =cd = a + b + c + d − 3.
Problem 22
Two circles have radii 15 and 95. If the two external tangents to the circles intersect at 60 degrees, how far apart are the centers of the circles?
Problem 23
A cubic block with dimensions n by n by n is made up of a collection of 1 by 1 by 1 unit cubes. What is the smallest value of n so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?
Problem 24

Problem 25
In the addition problem

each distinct letter represents a different digit. Find the number represented by the answer PICNIC.