EASY
IOQM - PRMO and RMO
IMPORTANT
Earn 100

There are 100 players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest 28 players are given a bye, and the remaining 72 players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. Find the total number of matches played.

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Important Questions on Counting

EASY
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 23
A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year 2003 ?
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 24
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many ways can a three-person planning committee be selected?
HARD
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 25
Find the least n such that whenever the elements of the set 1, 2, …, n are colored red or blue, there always exist x, y, z, w (not necessarily distinct) of the same color such that x+y+z=w.
HARD
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 26
There are n balls in a box, and the balls are numbered 1, 2, 3, …, n respectively. One of the balls is removed from the box, and it turns out that the sum of the numbers on the remaining balls in the box is 5048 . If the number on the ball removed from the box is m, find the value of m.
HARD
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 27
Consider the 800 -digit integer 234523452345⋯2345. The first m digits and the last n digits of the above integer are crossed out so that the sum of the remaining digits is 2345 . Find the value of m+n10.
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 28
The total number of integers between 0 and 105 having the digit sum equal to 8 is n, then find 5500-n.
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 29
How many ordered pairs of integers m,n, where 0<m<n<2008 satisfy the equation 20082+m2=20072+n2?
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 30
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?