Supported by
Charitable Trust
JMO mentoring scheme
March 2012 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 product of 4 consecutive odd numbers is 23961009. What is the smallest of these odd numbers?
2
Simplify Ö (164x ).
3
An equilateral triangle whose edges measure 20 cm is dissected and rearranged to form a square
without overlap or gaps. How long is the side of the square?
2
To see how this can be done, refer to www.mathworld.wolfram.com/Dissection.html
4
A 10 digit number abcdefghij includes all the digits 0 to 9. How many such numbers are there given
a + j = b + i = c + h = d + g = e + f = 9?
5
In a magic forest there are talking foxes, snakes and turtles. Turtles always speak the truth and foxes
always lie. Snakes only lie on rainy days. Hogwart’s student Hermione Granger talks to four of these
animals who tell her one by one: (1st) “it’s raining today”; (2nd) “that animal has just lied”;
(3rd) “the weather’s fair today”; (4th) “that animal has just lied or I'm a snake!”
At most, how many turtles did Hermione talk to?
6
Let x = m² - n², y = 2mn and z = m² + n² where m and n are integer KNs.
[See January question 7 and February question 6.]
Show that x - y = (m - n)² - 2n².
If x - y = 1 or y - x = 1, then [x, y, z] represents a Nearly Isosceles Right Angled Triangle [NIRAT],
namely a right angled triangle where the two shorter sides are nearly equal. Check using the KNs that
the right angled triangles with sides [3, 4, 5] and [20, 21, 29] fit these conditions.
Find the next largest NIRAT.
D OAB is equilateral. M is the mid-point of OA. A circle C1 has centre O and radius OA; only the arc
between A and B needs to be drawn. A second circle C2 has centre A and radius AM; only a quarter
circle needs to be drawn from M so that it cuts C1. Finally a circle C3 is formed with its centre N on
BM so that it touches both C1 and C2. Why does AN pass through the touching point of C2 and C3?
What line passes through the touching point of C1 and C3?
Let OA = 4. Show that the radius of C3 is 1.
If P is the intersection of C3 with NB and Q is the intersection of C3 with NM, show that MP = Ö 5 + 1
and that MP ´ MQ = 4.
7
3
4
5
6
7
Remember how to find the height of an equilateral triangle?
Remember that, once you've decided what a and j are, the other options are more limited.
You don't know what sort of day it was so consider the two cases of weather separately and count the possible turtles.
Note that [5, 12, 13] is not a NIRAT since one of the nearly equal sizes is z, the length of the hypotenuse.
Also be aware that x, y and z might not be in ascending order of magnitude.
You may find some insight by reflecting C1 in BM to produce another arc of radius 4 with centre A.
Draw the figure accurately though you need to know that the radius of C3 is 1 to construct N when you attempt it first time.