2013 PMWC

Problem I1

Solution

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

Solution

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

Solution

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

Solution

Problem I5

Solution

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

Problem I6

Solution

What is the 2013th term in the sequence

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

Problem I7

Solution

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

Which digit falls in the 100th place?

Problem I8

Solution

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

Solution

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

Problem I10

Solution

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

Solution

Problem I12

Solution

Problem I13

Solution

Problem I14

Solution

Problem I15

Solution

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

Problem T1

Solution

Problem T2

Solution

Problem T3

Solution

Problem T4

Solution

Problem T5

Solution

Problem T6

Solution

Problem T7

Solution

Problem T8

Solution

Problem T9

Solution

Problem T10

Solution