JMO mentoring scheme
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January 2013 paper
Generally earlier questions are easier and later questions more difficult.
Some questions are devised to help you learn aspects of mathematics which you may not meet in school.
Hints are upside down at the bottom of the page; fold the page back to view them when needed.
1
The number a = 1111…111 consists of 2013 digits, all 1s.
What is the remainder when a is divided by 37?
2
A teacher asks you to choose a number from 1 to 9, multiply it by 109, then to find the sum of the
digits in this product. Knowing the sum of the digits, the teacher is able to tell you with which number
you began. Explain how this can be done.
3
Alice, Brian and Carl have participated in a mathematics contest consisting of 12 problems. Before the
contest they were pessimistic and made the following statements :
Alice : ‘Brian will answer at least two more problems correctly than I will.’
Brian : ‘I will not answer more than five problems correctly.’
Carl : ‘I will not answer more problems correctly than Alice.’
Their teacher had also made a prediction that together they would answer more than 18 problems
correctly. Afterwards it transpired that all the students and the teacher had predicted incorrectly.
Who has/have answered the least number of problems correctly?
(A) only Alice ; (B) only Brian ; (C) only Carl ; (D) both Alice and Brian ; (E) not clear.
4
A (non-regular) heptagon ABCDEFG is given, all of whose sides have length 2. Ð DEF = 120 °,
Ð BCD = Ð FGA = 90 °, and Ð GAB = Ð ABC = Ð CDE = Ð EFG. What is the area of the heptagon?
5
Magminimus wasn't very good at arithmetic; he always got confused between numbers like XC and
CX. His father Magmaximus was an expert using arithmetic; he had become a very rich man, a senator
in Ancient Rome, and could afford to keep lots of slaves, including a very strict tutor for his son. The
tutor arranged 11 numbers around a circle and asked Magminimus to work out the differences between
adjacent numbers. (No-one knew about negative numbers at the time so simply subtracted the smaller
from the larger.) The boy gave 4 Is, 4 IIs and 3 IIIs as his answer. His tutor was quick to reproach him
disapprovingly; prove that Magminimus made a mistake.
6
A pointed arch ABC (like a Gothic arch) has a base AB of length 2.
The curved sections are arcs of circles constructed with centres A and B
and each with radius 2. A circle is inscribed inside the arch as shown.
What is the radius of this circle?
7
You are given two equations connecting p and r : p + pr + pr² = 28, p²r + p²r² + p²r³ = 224.
Prove that 2r² - 5r + 2 = 0.
Given this equation can be arranged as (r - 2)(2r - 1) = 0, find two solutions for p.
8
You are given a 7 by 7 grid of unit squares, 29 of which are black, the rest white.
Prove that a 2 by 2 square can be found containing at at least 3 black squares.
4
5
6
7
8
Dissect the figure into smaller shapes.
Think about odd and even numbers.
If O is the centre of the circle, show that the line joining O to the contact with the arc AC is also a radius from B to the arc.
Find the value of pr. For the second part, remember that if two numbers multiply to 0, then at least one of them is 0.
Try filling in as many 2 by 2 squares as you can and see what is left over.
How many positions can a 2 by 2 square be located on the grid?