# 2013 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

Nine cards are numbered from 1 to 9 respectively. Two cards are distributed to each of four children. The sum of the numbers on the two cards the children are given is: 7 for Ann, 10 for Ben, 11 for Cathy and 12 for Don. What is the number on the card that was not distributed?

## Problem I2

Given that *A*, *B*, *C* and *D* are distinct digits and

A A B C D - D A A B C = 2 0 1 3 D

Find A + B + C + D.

## Problem I3

A car traveled from Town A from Town B at an average speed of 100 km/h. It then traveled from Town B to Town C at an average speed of 75 km/h. Given that the distance from Town A to Town B is twice the distance from Town B to Town C, find the car's average speed, in km/h, for the entire journey.

## Problem I4

## Problem I5

Find the sum of all the digits in the integers from 1 to 2013.

## Problem I6

What is the 2013th term in the sequence

, , , , , , , , , , ...?

## Problem I7

All the perfect square numbers are written in order in a line: 14916253649...

Which digit falls in the 100th place?

## Problem I8

A team of four children are to be chosen from 3 girls and 6 boys. There must be at least one girl in the team. How many different teams of 4 are possible?

## Problem I9

The sum of 13 distinct positive integers is 2013. What is the maximum value of the smallest integer?

## Problem I10

Four teams participated in a soccer tournament. Each team played against all other teams exactly once. Three points were awarded for a win, one point for a draw and no points for a loss. At the end of the tournament, the four teams have obtained 5, 1, *x* and 6 points respectively. Find the value of *x*.

## Problem I11

## Problem I12

## Problem I13

## Problem I14

## Problem I15

Given that 1 + + + ... = *M* and 1 + + + ... = *K*, find the ratio of M : K .