Maths Olympiad

smiles

Blow your brains out - i did.

Finals of the Irish Maths Olympiad today.
Two 3 hour papers. Pencil and paper only. (no calculators!)

Paper I:
1 - In a triangle ABC. AB = 20, AC = 21, and BC = 29. The points D and E lie on the line segment BC, with BD = 8 and EC = 9. Calculate the angle DAE.

2 - (a) A group of people attend a part. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a mutual acquaintance in the group. What is the maximum number of people present?

(b) If, in addition, the group contains three mutual acquaintances (ie. three people each of whom knows the other two), what is the maximum number of people?

3 - Find all triples of positive integers (p, q, n), with p and q primes satisfying:

p(p+3) + q(q+3) = n(n+3)

The `number is meant to be subscript
4 - Let the sequence a`1, a`2, a`3, a`4... be defined by:
a`1 = 1, a`2 = 1, a`3 = 1.
and
[a`(n+1)][a`(n-2)] - [a`n][a`(n-1)] = 2,
for all n >= 3. Prove that a`n is a positive integer.

the 3(abc)^1/3 is 3 multiplied by the cube-root of abc
5 - Let 0 < a, b, c < 1. Prove that:

a/(1-a) + b/(1-b) + c/(1-c) >=[ 3(abc)^1/3]/[1 - (abc)^1/3]

Also: Determine the case of equality.


lovely!?

<< Fio >>

sisob

ouch

smiles

6 - A (3 x n) grid is filled as follows: the first row consists of the numbers from 1 to n arranged from left to right in ascending order. The second row is a cyclic shift of the top row. Thus the order goes i, i+1, ... , n-1, n, n+1, ... , i - 1 for some i. The third row has the numbers 1 to n in some order, subject to the rule that in each of the n columns, the sum of the three numbers is the same.

For which values of n is it posssible to fill the grid according to the above rules? For an n for which this is possible, determine the number of different ways of filling the grid.

7 - Support n is a product of four distinct primes a, b, c, d such that:

(a) a + c = d;
(b) a(a + b + c +d) = c(d - b);
(c) 1 + bc + d = bd.

Determine n.

8 - Denote by Q the set of rational numbers. Determine all functions f:Q -> Q such that:

f(x + f(y)) = y + f(x), for all x, y elements of Q.

9 - For each real numer x, define {x} to be the greatest integer less than or equal to x.
Let A = 2 + (squareroot of 3). Prove that:

(A^n) - {(A^n)} = 1 - (A^-n), for n = 0, 1, 2, 3, ...

10 - Let ABC be a triange whose side lengths are all integers, and let D and E be the points at which the incircle of ABC touches BC and AC respectively.
If | (AD)^2 - (BE)^2 | <= 2, show that AC = BC.


Have fun! :P

<< Fio >>

Man U babe

Mein Gott.
Did maths olympiad 2 years ago, didnt get past prelims in UCD. Kinda wish I did it this year but theres no point regretting now. Think I would have taken the "nod and smile" approach to that one though!
How did ya get on anyway?