# 2000 PMWC Problems

## Contents

- 1 Problem I1
- 2 Problem I2
- 3 Problem I3
- 4 Problem I4
- 5 Problem I5
- 6 Problem I6
- 7 Problem I7
- 8 Problem I8
- 9 Problem I9
- 10 Problem I10
- 11 Problem I11
- 12 Problem I12
- 13 Problem I13
- 14 Problem I14
- 15 Problem I15
- 16 Problem T1
- 17 Problem T2
- 18 Problem T3
- 19 Problem T4
- 20 Problem T5
- 21 Problem T6
- 22 Problem T7
- 23 Problem T8
- 24 Problem T9
- 25 Problem T10

## Problem I1

is a number that has different factors (including the number and itself). What is the smallest possible value of ?

## Problem I2

As far as we know, the greatest prime number is . What is the remainder when is divided by ?

## Problem I3

How many whole numbers less than contain at least one but no ?

## Problem I4

Given that . If and are positive integers, find the smallest value of .

## Problem I5

In a language college, students can speak Chinese, students can speak English, and students can speak neither Chinese nor English. Find the percentage of students who can speak both Chinese and English.

## Problem I6

is the product of two -digit numbers formed by the digits without any repetition. Find the largest value of in the form of . (You are not required to multiply the numbers).

## Problem I7

and are two numbers that have prime factors and only. has factors ( and itself are included), has factors ( and itself are included), and their HCF (Highest Common Factor) is . What is the LCM (Least Common Multiple of and ?

## Problem I8

A circle and a triangle are placed on intersections of the grid. The circle and the triangle are not allowed to lie on the same vertical and horizontal line. How many total possible ways are there of placing the circle and the triangle? (the following is an example).

## Problem I9

Using only odd digits, all possible three-digit numbers are formed. Determine the sum of all such numbers.

## Problem I10

In the sum , each letter represents a distinct digit from to . The sum is as close as possible to without being greater than or equal to . What is the sum?

## Problem I11

You have a pack of cards, among which are red, yellow, green and blue ones. How many cards would you need to draw out to ensure that you have cards of the same colour?

## Problem I12

During the rest hour, one of five students (, , , , and ) dropped a glass of water. The following are the responses of the children when the teacher questioned them:

- : It was or who dropped it.
- : Neither nor I did it.
- : Both and are lying.
- : Only one of or is telling the truth.
- : is not speaking the truth.

The class teacher knows that three of them NEVER lie while the other two ALWAYS lie. Who dropped the glass?

## Problem I13

In the figure, the squares and both have the same area of . is a semicircle. The point is the mid-point of the arc . Find the area of the shaded part. (Assume )

## Problem I14

A copy machine has a following enlargement/reduction buttons:

The buttons , , and are out of order and cannot be used any more. Sam wants to make a copy that is the same size as the original document by using the remaining buttons. When he presses a button, he has to pay . What is the minimum amount he has to pay?

## Problem I15

The sum of several positive integers is . Find the largest product that can be formed by these integers.

## Problem T1

A box contains to candies. When the candies are evenly distributed to , , , , or children, there is always one candy left. If the candies are packaged into small bags each having the same number of candies, what is the largest number of candies below in each bag so that no candies are left?

## Problem T2

A cube is formed using cubes. A number of the smaller cubes are removed by punching out the 15 designated columns from front to back, top to bottom, and side to side. Find the number of smaller cubes that remain.

## Problem T3

In the figure, is a square, , , , and are midpoints of the sides , , and respectively. Find the ratio of the shaded area to the area of the square .

## Problem T4

Use the four colours red, yellow, blue and green to fill in the regions of the following diagram so that the adjacent regions are not the same colour. How many different ways are there to colour the regions?

## Problem T5

Each of children ranked kinds of chocolate (A, B, and C) in order of their preferences, with no absentions. Suppose children preferred B to C, preferred C to A, and preferred A to B, and all possible ordered of A, B, and C existed. Find the number of children who ranked A as their first preference.

## Problem T6

In , , , , . Find the ratio of the area of to the area of .

## Problem T7

The digits to without repetition form two 5-digit numbers and . Given that equals to , find the largest possible sum of and .

## Problem T8

There are positive integers , , such that . What is the smallest possible value of ?