Zimaths competition

There are two sections in this competition: normal is open to any high school student, abnormal is open to anyone, teachers included. The hardness of questions will increase as readers become familiar with more and more concepts. If you have any questions to send us, please do. Regrettably, no prizes are available to readers outside Zimbabwe.

Click here for the previous contest.

Normal Section

1. Tortoise takes two minutes to walk from his house to the drinking pool. Hare also starts moving at the same time, running repeatedly from the pool to Tortoise and back, till Tortoise reaches the pool. If it takes Hare five minutes to run a kilometre, how far does he run?
2. How many square numbers (1, 4, 9, 16, etc) are there which are one greater than a positive prime number?
3. In an acute-angled triangle, each angle is an whole no. of degrees and the smallest angle is a fifth of the largest angle. Find all the triangle's angles. (International Tournament of the Towns, 1996)

Abnormal Section

1. What's the next number in the series
   
   4 , 5 , 2 2/9 , 1 7/13 , 10/11 , 4/7 ,...?
2. Show that the product of 4 consecutive natural numbers can never be a square number.
3. Consider the sum

   		P	A	R	M	A
   		V	I	L 	L	A
   	+	B	A	R	C	A
   	------------------------------------------
   	M	O	N	A	C	O
   	------------------------------------------
   

   In the above sum, each letter stands for a single digit. Digits are 0,1,\ldots,9. The values for the vowels are such that A < I < O, and these values cannot be held by any other letter. If P, R, I and M represent prime numbers, show that this sum is impossible.
4. A new soccer league has been proposed with over 70 teams. Each team plays each other at least once. Each match is played on one of four types of pitch --- sand, grass, astroturf or mud. Show that you will always find three teams who play each other on one type of pitch.

In the UZ staff canteen's kitchen, 8 labelled jars can always be found on the kitchen shelf, namely those for raisins, rice, barley, salt, sugar, flour, currants and tea. But things have been rather chaotic of late, and now exactly seven of the eight jars hold the wrong commodity. Given the following information, tell us what's in each container.

Neither salt nor sugar is in the tea-jar, nor is the tea in either of their jars. The rice-jar holds no rice, barley or currants. The barley-jar holds no rice, and the currant-jar holds no barley. The contents of the sugar-jar should be where the barley is, and the currant-jar holds what ought to be where the flour is, which is not in the tea-jar. Sugar is not in the salt-jar, nor salt in the sugar-jar, nor barley in the flour-jar. The label on the jar holding the sugar describes the contents of the raisin-jar. The rice is neither in the tea-jar nor the salt-jar. The salt is not in the flour-jar, and no two items are both in each other's jar.

Click here for the questions in the next contest.