Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2013 PMWC

Revision as of 07:45, 3 January 2014 by Sarcasticpie (talk | contribs) (→‎Problem I10)

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

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

Problem I7

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

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2013_PMWC&oldid=58569"