The 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. The questions of our sample issue (not on the web) have also been included. If you have any questions to send us, please do. Regrettably, no prizes are available to readers outside Zimbabwe.

Questions of the sample issue

Normal section

1. The teacher had to give 6 books back to 6 students after a test (each student must get a book). In how many ways can he give just 1 book to the wrong owner and the rest to their right owners?
2. A political party has 2000 affiliates. At a congress, 12.12121212121212...% of those present are women and 3.4234234234234234...% of those present are radicals. How many delegates are absent?
3. Two trains are approaching each other on a straight line, one at 24m/s and another at 30m/s. If they are initially 100km apart, find out how far apart they are a minute before passing each other.

Abnormal competition

1. The product of two of the four roots of the quartic equation x⁴-18x³+kx²+200x=1984 is -32. Find k.
2. Say one picks three random numbers each between 0 and 1. What is the probability that their sum, when rounded to the nearest integer, is 1?
3. Prove that . Note that [x] refers to the largest integer less than or equal to x, eg [4]=4, [10.43]=10 and [-3.4] = -4.

This issue's questions

Normal Section

1. Find the exact difference between a⁶ and the nearest integer, where a is the golden ratio .
2. Shown is a variation of the Penrose staircase, developed by the British mathematician Roger Penrose. It is required to get from E to A by climbing the least number of steps. How can this be done? Don't worry about the varying sizes of the steps, we are just interested in the number you have to climb. (As for the diagram, it's really terrible, but just assume all the faces that are horizontallish are really horizontal)
3. Prove that the last digit of 1 + 2 + 3 + ... + n (where n is a positive integer) cannot be 2,4,7 or 9.
4. Consider the 100th row of Pascal's Triangle, the one that goes like 1, 100, 4950, ... , 4950, 100, 1. Is the number of its odd entries even or odd?

Abnormal Section

1. Prove that 7⁷⁷⁷⁷-1 is divisible by six.
2. A point P is taken on the curve y=x³. The tangent at P meets the curve again at Q. Prove that the slope of the curve at Q is 4 times the slope of the curve at P.
3. A sum of two three digit integers, where the digits of one are in the opposite order to the digits of the other, is 1252. Find these integers if the sum of the digits of either of them is 14 and the sum of the squares of the same digits is 84.
4. Prove that