According to the organizers, the key competence tested by Mathematical Kangaroo is logical combination, not just pure knowledge of formulas.Perimeter and area of a square, a rectangle The competition was established in 1991 by André Deledicq, a professor of mathematics at the University of Paris 7, and Jean-Pierre Boudine, professor of mathematics at Marseille.

Here are a few examples of course problems with their solutions.

**Question** The positive integer mm has leading digit 1. When this digit is moved to the other end, the result is 3m. What is the smallest such m?

**Answer** It is useful to find what happens when the digits from 0 to 9 are multiplied by 3. the tables show the resulting last digits.
we know that 3m ends in a 1. from the table, the only digit that can be multiplied by 3 to give a units digit of 1 is 7. thus m ends in 7 and so 3m ends in 71.
hence we now wish to find a two-digit number 'a7' which can be multiplied by 3 to give something ending in 71. sine 3 x 7 = 21 and 71 - 21= 50, we require 3 x a to end in 5, and from the table we deduce that a =5. thus m ends in 57, and 3 x 57= 171.
Similarly, when we look for a four-digit number ending in 857 which can be multiplied by 3 o give something ending in 8571, we find only 2857, and 3 x 2857= 8571. when we look for a five-digit number ending in 2857 which can be multiplied by 3 to give something ending in 28571, we find 3 x 42 857 = 428 571.
so m is 142 857, and indeed when we move the 1 to the end we get
428 571 = 3 x 1 42 857.

**Problem** Pablo plans to take several unit cubes and arrange them to form a larger cube. He will then paint some of the faces of the larger cube. When the paint has dried, he will split the larger cube into unit cubes again. Suppose that Pablo wants exactly 150 of the unit cubes to have no paint on them at all. How many faces of the larger cube should he paint?

**Solution** Suppose that the larger cube is made up of n x n x n smaller cubes. now the unpainted smaller cubes from a cuboid, and we can work out the number of unpainted smaller cubes in the form abc. here a is n (if neither the left nor the right face is painted), or n - 1 (if exactly one of the two faces is painted), or n - 2 (if both are). similarly, b is n, n - 1 or n - 2 depending on which of the front and back faces are painted.
so we are trying to express 150 as a product of three numbers that differ by at most 2. now the prime factorization of 150 is is 2 x 3 x 5 x . combining the 2 and 3 gives 150= 5 x 5 x 6, which is of the desired form. any other option has the form 150= 1 x r x s, or 150= 2 x r x s or 150= 3 x r x s, and none of these is possible since one of r and s will differ from the first factor by more than 2
finally, abc= 5 x 5 x 6 corresponds to eiher
abc=(n-1)x(n-1)xn with n = 6
or abc=(n-2)x(n-2)x(n-1) with n=7.

**Asking**A mathematician has a full one-litre bottle of concentrated orange squash, a large container and a tap. He first pours half of the bottle of orange squash into the container. Then he fills the bottle from the tap, shakes well, and pours half of the resulting mixture into the container. He then repeats this step over and over again: filling the bottle from the tap each time, shaking the mixture well, and then pouring half of the contents into the container. Suppose that on the final occasion he fills the bottle from the tap and empties it completely into the container. How many times has he filled the bottle from the tap if the final mixture consists of 10% orange squash concentrate?

**Response** 18

The idea comes from the Australian Mathematics Competition, initiated in 1978 by Peter O'Halloran.Addition, subtraction, multiplication, division. intersection of sets. According to the organizers, the key competence tested by Mathematical Kangaroo is logical combination, not just pure knowledge of formulas.Simple arithmetic operations with 1,2,3 and 4-digit numbers

Cayley

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Grey

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Hamilton

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IMC

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Exam

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JMC

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JMO

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Pink

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SMC

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