Understand the problem

Find the 3-digit number whose ratio with the sum of its digits is minimal.

Source of the problem

Albania TST 2013

Topic
Number Theory, Inequalities.
Difficulty Level
Easy
Suggested Book
Problem Solving Strategies by Arthur Engel

Start with hints

Do you really need a hint? Try it first!

Suppose that the number is (abc)_{10}. Then we would like to minimise \frac{100a+10b+c}{a+b+c}. Try to minimise for one variable at a time.
Note that, \frac{100a+10b+c}{a+b+c}=1+\frac{9(11a+b)}{a+b+c}. In this expression, it is possible to minimise for c independent of a,b.
From the previous hint, the expression is minimised for c=9. Show that the ratio can now be rewritten as 1+\frac{10a-9}{a+b+9}. Minimise for $b$.
In the previous hint we see that the minimising value of b is also 9. Finally, the ratio may be written as 10-\frac{189}{a+18}. This is minimised for a=1. Hence the answer is 199.

Watch the video (Coming Soon)

Connected Program at Cheenta

Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Similar Problems

Lines and Angles | PRMO 2019 | Question 7

Try this beautiful problem from the Pre-RMO, 2019 based on Lines and Angles. You may use sequential hints to solve the problem.

Logarithm and Equations | AIME I, 2012 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Logarithm and Equations.

Cross section of solids and volumes | AIME I 2012 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Cross section of solids and volumes.

Angles of Star | AMC 8, 2000 | Problem 24

Try this beautiful problem from GeometryAMC-8, 2000 ,Problem-24, based triangle. You may use sequential hints to solve the problem.

Unit digit | Algebra | AMC 8, 2014 | Problem 22

Try this beautiful problem from Algebra about unit digit from AMC-8, 2014. You may use sequential hints to solve the problem.

Problem based on Integer | PRMO-2018 | Problem 6

Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.

Number counting | ISI-B.stat Entrance | Objective from TOMATO

Try this beautiful problem Based on Number counting .You may use sequential hints to solve the problem.

Area of a Triangle | AMC-8, 2000 | Problem 25

Try this beautiful problem from Geometry: Area of the triangle from AMC-8, 2000, Problem-25. You may use sequential hints to solve the problem.

Trapezium | Geometry | PRMO-2018 | Problem 5

Try this beautiful problem from Geometry based on Trapezium from PRMO , 2018. You may use sequential hints to solve the problem.

Probability Problem | AMC 8, 2016 | Problem no. 21

Try this beautiful problem from Probability from AMC-8, 2016 Problem 21. You may use sequential hints to solve the problem.