MathDB

1989 Irish Math Olympiad

Part of Ireland National Math Olympiad

Subcontests

(5)
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Inequality on a Quadrilateral

A quadrilateral ABCDABCD is inscribed, as shown, in a square of area one unit. Prove that 2≤∣AB∣2+∣BC∣2+∣CD∣2+∣DA∣2≤42\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4 [asy] size(6cm); draw((0,0)--(10,0)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((0,0)--(0,10));
dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0)); label("DD",(0,8.5),W); label("AA",(3.5,10),NE); label("BB",(10,3.5),E); label("CC",(3.5,0),S); draw((0,8.5)--(3.5,10)); draw((3.5,10)--(10,3.5)); draw((10,3.5)--(3.5,0)); draw((3.5,0)--(0,8.5)); [/asy]

Locus of Q as P Varies Along L

Suppose LL is a fixed line, and AA is a fixed point not on LL. Let kk be a fixed nonzero real number. For PP a point on LL, let QQ be a point on the line APAP with ∣AP∣⋅∣AQ∣=k2|AP|\cdot |AQ|=k^2. Determine the locus of QQ as PP varies along the line LL.
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Ireland Olympiad number theory

Let x=a1a2…anx = a_1a_2 \dots a_n be an n-digit number, where a1,a2,…,an(a1≠0)a_1, a_2, \dots , an (a_1 \neq 0) are the digits. The nn numbers x1=x=a1a2...an, x_1 = x = a_1 a_2 ... a_n, x2=ana1...an−1, x_2 = a_n a_1 ... a_{n-1}, x3=an−1ana1...an−2 x_3 = a_{n-1} a_n a _1 ... a_{n-2} , x4=an−2an−1ana1,...an−3, x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} , ...,xn=a2a3...ana1 ... , x_n = a_2 a_3 ... a_n a_1 are said to be obtained from xx by the cyclic permutation of digits. [For example, if n=5n = 5 and x=37001x = 37001, then the numbers are x1=37001,x2=13700,x_1 = 37001, x_2 = 13700, x3=01370(=1370),x4=00137(=137),x_3 = 01370(= 1370), x_4 = 00137(= 137), x5=70013.] x_5 = 70013.] Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.

Bertrand and Prime Counting

(i): Prove that if nn is a positive integer, then (2nn)=(2n)!(n!)2\binom{2n}{n}=\frac{(2n)!}{(n!)^2} is a positive integer that is divisible by all prime numbers pp with n<p≤2nn<p\le 2n, and that (2nn)<22n.\binom{2n}{n}<2^{2n}.
(ii): For xx a positive real number, let π(x)\pi(x) denote the number of prime numbers p≤xp \le x. [Thus, π(10)=4\pi(10) = 4 since there are 44 primes, viz., 22, 33, 55, and 77, not exceeding 1010.]Prove that if n≥3n \ge 3 is an integer, then (a)π(2n)<π(n)+2nlog⁡2(n);\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}};(b)π(2n)<2n+1log⁡2(n−1)n;\pi(2^n) < {{2^{n+1}\log_2(n-1)}\over{n}};(c) Deduce that, for all real numbers x≥8x \ge 8,π(x)<4xlog⁡2(log⁡2(x))log⁡2(x).\pi(x) < {{4x \log_2(\log_2(x))}\over{\log_2(x)}}.
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Maximum Value of Function in Range

A function ff is defined on the natural numbers N\mathbb{N} and satisfies the following rules:
(a) f(1)=1f(1)=1;
(b) f(2n)=f(n)f(2n)=f(n) and f(2n+1)=f(2n)+1f(2n+1)=f(2n)+1 for all n∈Nn\in \mathbb{N}.
Calculate the maximum value mm of the set {f(n):n∈N,1≤n≤1989}\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}, and determine the number of natural numbers nn, with 1≤n≤19891\le n\le 1989, that satisfy the equation f(n)=mf(n)=m.

Geometry Triangle inequality

Suppose P is a point in the interior of a triangle ABC, that x; y; z are the distances from P to A; B; C, respectively, and that p; q; r are the per- pendicular distances from P to the sides BC; CA; AB, respectively. Prove that xyz \geq 8pqr; with equality implying that the triangle ABC is equilateral.
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3x3 Magic Square

A 3x3 magic square, with magic number mm, is a 3×33\times 3 matrix such that the entries on each row, each column and each diagonal sum to mm. Show that if the square has positive integer entries, then mm is divisible by 33, and each entry of the square is at most 2n−12n-1, where m=3nm=3n. An example of a magic square with m=6m=6 is
(213321132)\left( \begin{array}{ccccc} 2 & 1 & 3\\ 3 & 2 & 1\\ 1 & 3 & 2 \end{array} \right)

Information Passing problem

2. Each of nn members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.
Hi, from my sketches I'm thinking the answer is 2n−22n-2 but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks