# 2013 PMWC Problems

## 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 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 $\frac{1}{1}$ , $\frac{2}{1}$ , $\frac{1}{2}$ , $\frac{3}{1}$ , $\frac{2}{2}$ , $\frac{1}{3}$ , $\frac{4}{1}$ , $\frac{3}{2}$ , $\frac{2}{3}$ , $\frac{1}{4}$ , ...?

## 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 I15

Given that 1 + $\frac{1}{2^2}$ + $\frac{1}{3^2}$ + ... = M and 1 + $\frac{1}{3^2}$ + $\frac{1}{5^2}$ + ... = K, find the ratio of M : K .