Navvye/Olympiad-Problems-CSV huggingface dataset:
1996 Iran MO (3rd Round)
Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that

$$\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.$$
1996 Iran MO (3rd Round)
Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.
1996 Iran MO (3rd Round)
Consider a semicircle of center $O$ and diameter $AB$. A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ s.t. $MC>MD$ and $MB<MA$. The circumcircles od the $AOC$ and $BOD$ intersect again at $K$. Prove that $MK\perp KO$.
1996 Iran MO (3rd Round)
Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.
1996 Iran MO (3rd Round)
Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that

$$f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.$$
1996 Iran MO (3rd Round)
Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.
1996 Iran MO (3rd Round)
Find all pairs $(p,q)$ of prime numbers such that

$$m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.$$
1993 Iran MO (3rd Round)
Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.
1993 Iran MO (3rd Round)
In the figure below, area of triangles $AOD, DOC,$ and $AOB$ is given. Find the area of triangle $OEF$ in terms of area of these three triangles.



[asy]

import graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,ymin=-6.6,ymax=5.16; 

pair A=(0,0), F=(9,0), B=(4,0), C=(3.5,2), D=(1.94,2.59), O=(2.75,1.57); 

draw(A--(3,4),linewidth(1.2)); draw((3,4)--F,linewidth(1.2)); draw(A--F,linewidth(1.2)); draw((3,4)--B,linewidth(1.2)); draw(A--C,linewidth(1.2)); draw(B--D,linewidth(1.2)); draw((3,4)--O,linewidth(1.2)); draw(C--F,linewidth(1.2)); draw(F--O,linewidth(1.2)); 

dot(A,ds); label("$A$",(-0.28,-0.23),NE*lsf); dot(F,ds); label("$F$",(8.79,-0.4),NE*lsf); dot((3,4),ds); label("$E$",(3.05,4.08),NE*lsf); dot(B,ds); label("$B$",(4.05,0.09),NE*lsf); dot(C,ds); label("$C$",(3.55,2.08),NE*lsf); dot(D,ds); label("$D$",(1.76,2.71),NE*lsf); dot(O,ds); label("$O$",(2.57,1.17),NE*lsf); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
1993 Iran MO (3rd Round)
Prove that there exists a subset $S$ of positive integers such that we can represent each positive integer as difference of two elements of $S$ in exactly one way.
1993 Iran MO (3rd Round)
In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal. We construct four equilateral triangles with centers $O_1,O_2,O_3,O_4$ on the sides sides $AB, BC, CD, DA$ outside of this quadrilateral, respectively. Show that $O_1O_3 \perp O_2O_4$.
1993 Iran MO (3rd Round)
Let $x_1, x_2, \ldots, x_{12}$ be twelve real numbers such that for each $1 \leq i \leq 12$, we have $|x_i| \geq 1$. Let $I=[a,b]$ be an interval such that $b-a \leq 2$. Prove that number of the numbers of the form $t= \sum_{i=1}^{12} r_ix_i$, where $r_i=\pm 1$, which lie inside the interval $I$, is less than $1000$.
1988 Irish Math Olympiad
A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.
1988 Irish Math Olympiad
A; B; C; D are the vertices of a square, and P is a point on the arc CD of

its circumcircle. Prove that

$ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $

Can anyone here find the solution? I'm not great with geometry, so i tried turning it into co-ordinate geometry equations, but sadly to no avail. Thanks in advance.
1988 Irish Math Olympiad
$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.
1988 Irish Math Olympiad
Problem: A mathematical moron is given the values b; c; A for a triangle ABC and

is required to find a. He does this by using the cosine rule

$ a^2 = b^2 + c^2 - 2bccosA$

and misapplying the low of the logarithm to this to get

$ log a^2 = log b^2 + log c^2 - log(2bc cos A) $

He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms

and gets the correct answer. What can be said about the triangle ABC?
1988 Irish Math Olympiad
Problem: A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). In how many ways can this be done so that all friends are invited at least once?
1988 Irish Math Olympiad
Suppose you are given $n$ blocks, each of which weighs an integral number of pounds, but less than $n$ pounds. Suppose also that the total weight of the $n$ blocks is less than $2n$ pounds. Prove that the blocks can be divided into two groups, one of which weighs exactly $n$ pounds.
1988 Irish Math Olympiad
A function $f$, defined on the set of real numbers $\mathbb{R}$ is said to have a horizontal chord of length $a>0$ if there is a real number $x$ such that $f(a+x)=f(x)$. Show that the cubic $$f(x)=x^3-x\quad \quad \quad \quad (x\in \mathbb{R})$$has a horizontal chord of length $a$ if, and only if, $0<a\le 2$.
1988 Irish Math Olympiad
Let $x_1,x_2,x_3,\dots$ be sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}, \quad \quad n=3,4,5,\dots$$Establish necessary and sufficient conditions on $x_1,x_2$ for $x_n$ to be an integer for infinitely many values of $n$.
1988 Irish Math Olympiad
The year $1978$ was “peculiar” in that the sum of the numbers formed with the first two digits and the last two digits is equal to the number formed with the middle two digits, i.e., $19+78=97$. What was the last previous peculiar year, and when will the next one occur?
1988 Irish Math Olympiad
Let $0\le x\le 1$. Show that if $n$ is any positive integer, then $$(1+x)^n\ge (1-x)^n+2nx(1-x^2)^{\frac{n-1}{2}}$$.
1988 Irish Math Olympiad
If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $1/a$, $a>0$, to use the iteration $$x_{k+1}=x_k(2-ax_k), \quad \quad k=0,1,2,\dots ,$$where $x_0$ is a selected “starting” value. Find the limitations, if any, on the starting values $x_0$, in order that the above iteration converges to the desired value $1/a$.
1988 Irish Math Olympiad
Prove that if $n$ is a positive integer ,then $$cos^4\frac{\pi}{2n+1}+cos^4\frac{2\pi}{2n+1}+\cdots+cos^4\frac{n\pi}{2n+1}=\frac{6n-5}{16}.$$
1988 Irish Math Olympiad
The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold:

(a) $E$ is the mid-point of $AB$;

(b) points $A,G$ and $F$ are on the same line;

(c) there is a point $C$ at which $BG$ and $EF$ intersect;

(d) $|CE|=1$ and $|AC|=|AE|=|FG|$.

Show that if $|AG|=x$, then $|AB|=x^3$.
1988 Irish Math Olympiad
2. Let $x_1, . . . , x_n$ be $n$ integers, and let $p$ be a positive integer, with $p < n$. Put

$$S_1 = x_1 + x_2 + . . . + x_p$$$$T_1 = x_{p+1} + x_{p+2} + . . . + x_n$$$$S_2 = x_2 + x_3 + . . . + x_{p+1}$$$$T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1$$$$...$$$$S_n=x_n+x_1+...+x_{p-1}$$$$T_n=x_p+x_{p+1}+...+x_{n-1}$$For $a = 0, 1, 2, 3$, and $b = 0, 1, 2, 3$, let $m(a, b)$ be the number of numbers $i$, $1 \leq i \leq n$, such that $S_i$ leaves remainder $a$ on division by $4$ and $T_i$ leaves remainder $b$ on division by $4$. Show that $m(1, 3)$ and $m(3, 1)$ leave the same remainder when divided by $4$ if, and only if, $m(2, 2)$ is even.
1988 Irish Math Olympiad
A city has a system of bus routes laid out in such a way that

(a) there are exactly $11$ bus stops on each route;

(b) it is possible to travel between any two bus stops without changing routes;

(c) any two bus routes have exactly one bus stop in common.

What is the number of bus routes in the city?
1989 Irish Math Olympiad
A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$[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("$D$",(0,8.5),W);

label("$A$",(3.5,10),NE);

label("$B$",(10,3.5),E);

label("$C$",(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]
1989 Irish Math Olympiad
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is



$$\left( \begin{array}{ccccc}

2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2

\end{array} \right)$$
1989 Irish Math Olympiad
A function $f$ is defined on the natural numbers $\mathbb{N}$ and satisfies the following rules:



(a) $f(1)=1$;



(b) $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$ for all $n\in \mathbb{N}$.



Calculate the maximum value $m$ of the set $\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}$, and determine the number of natural numbers $n$, with $1\le n\le 1989$, that satisfy the equation $f(n)=m$.
1989 Irish Math Olympiad
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.
1989 Irish Math Olympiad
Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq  0)$ are the digits. The $n$ numbers $ x_1 = x = a_1 a_2 ... a_n, $ $ x_2 = a_n a_1 ... a_{n-1}, $ $ x_3 = a_{n-1} a_n a _1 ... a_{n-2} $ ,

$ x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} , $ $ ... , x_n = a_2 a_3 ... a_n a_1$

are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$, then the numbers are $x_1 = 37001, x_2 = 13700, $ $x_3 = 01370(= 1370), x_4 = 00137(= 137), $ $ 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.
1989 Irish Math Olympiad
Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.
1989 Irish Math Olympiad
2. Each of  $n$ 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 $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks
1989 Irish Math Olympiad
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.
1989 Irish Math Olympiad
Let $a$ be a positive real number and let



$b= \sqrt[3] {a+ \sqrt {a^{2}+1}} +  \sqrt[3] {a- \sqrt {a^{2}+1}}$.



Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.
1989 Irish Math Olympiad
(i): Prove that if $n$ is a positive integer, then $$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$$is a positive integer that is divisible by all prime numbers $p$ with $n<p\le 2n$, and that $$\binom{2n}{n}<2^{2n}.$$

(ii): For $x$ a positive real number, let $\pi(x)$ denote the number of prime numbers $p \le x$. [Thus, $\pi(10) = 4$ since there are $4$ primes, viz., $2$, $3$, $5$, and $7$, not exceeding $10$.]Prove that if $n \ge 3$ is an integer, then

(a)$$\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}};$$(b)$$\pi(2^n) < {{2^{n+1}\log_2(n-1)}\over{n}};$$(c) Deduce that, for all real numbers $x \ge 8$,$$\pi(x) < {{4x \log_2(\log_2(x))}\over{\log_2(x)}}.$$
1990 Irish Math Olympiad
Given a natural number $n$, calculate the number of rectangles in the plane, the coordinates of whose vertices are integers in the range $0$ to $n$, and whose sides are parallel to the axes.
1990 Irish Math Olympiad
A sequence of primes $a_n$ is defined as follows: $a_1 = 2$, and, for all $n \geq 2$,$

a_n$ is the largest prime divisor of $a_1a_2...a_{n-1} + 1$. Prove that $a_n \neq 5$

for all n.

I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks
1990 Irish Math Olympiad
Determine whether there exists a function $ f: \mathbb{N}\longrightarrow \mathbb{N}$ such that

$ f(n)=f(f(n-1))+f(f(n+1))$ for all natural numbers $ n\ge 2$.
1990 Irish Math Olympiad
The real number $x$ satisfies all the inequalities $$2^k<x^k+x^{k+1}<2^{k+1}$$for $k=1,2,\dots ,n$. What is the greatest possible value of $n$?
1990 Irish Math Olympiad
Let $ABC$ be a right-angled triangle with right-angle at $A$. Let $X$ be the foot of the perpendicular from $A$ to $BC$, and $Y$ the mid-point of $XC$. Let $AB$ be extended to $D$ so that $|AB|=|BD|$. Prove that $DX$ is perpendicular to $AY$.
1990 Irish Math Olympiad
Let $n$ be a natural number, and suppose that the equation $$x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0$$has a solution with all the $x_i$s equal to $\pm 1$. Prove that $n$ is divisible by $4$.
1990 Irish Math Olympiad
Let $n>3$ be a natural number . Prove that $$\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.$$
1990 Irish Math Olympiad
Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.
1990 Irish Math Olympiad
Let $t$ be a real number, and let $$a_n=2\cos \left(\frac{t}{2^n}\right)-1,\quad n=1,2,3,\dots$$Let $b_n$ be the product $a_1a_2a_3\cdots a_n$. Find a formula for $b_n$ that does not involve a product of $n$ terms, and deduce that $$\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}$$
1990 Irish Math Olympiad
Let $n=2k-1$, where $k\ge 6$ is an integer. Let $T$ be the set of all $n$-tuples $$\textbf{x}=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.$$For $\textbf{x}=(x_1,x_2,\dots ,x_n)$ and $\textbf{y}=(y_1,y_2,\dots ,y_n)$ in $T$, let $d(\textbf{x},\textbf{y})$ denote the number of integers $j$ with $1\le j\le n$ such that $x_j\neq x_y$. $($In particular,  $d(\textbf{x},\textbf{x})=0)$.



Suppose that there exists a subset $S$ of $T$ with $2^k$ elements which has the following property: given any element $\textbf{x}$ in $T$, there is a unique $\textbf{y}$ in $S$ with  $d(\textbf{x},\textbf{y})\le 3$.



Prove that $n=23$.
1991 Irish Math Olympiad
Three points $X,Y$ and $Z$ are given that are, respectively, the circumcenter of a triangle $ABC$, the mid-point of $BC$, and the foot of the altitude from $B$ on $AC$. Show how to reconstruct the triangle $ABC$.
1991 Irish Math Olympiad
Problem:

Find all polynomials satisfying the equation

$ f(x^2) = (f(x))^2 $

for all real numbers x.

I'm not exactly sure where to start though it doesn't look too difficult. Thanks!
1991 Irish Math Olympiad
Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows:

$f(n)=10n$, if $n$ is a positive integer;

$g(n)=10n+4$, if $n$ is a positive integer;

$h(n)=\frac{n}{2}$, if $n$ is an even positive integer.

Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order.



$[$For example: $35=h(f(h(g(h(h(4)))))).]$
1991 Irish Math Olympiad
Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament.

They decided on the following rules of attendance:



(a) There should always be at least one person present on each day.



(b) On no two days should the same subset attend.



(c) The members present on day $N$ should include for each $K<N$, $(K\ge 1)$ at least one member who was present on day $K$.



For how many days can the parliament sit before one of the rules is broken?
1991 Irish Math Olympiad
Find all polynomials



$f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$



with the following properties



(a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and

(b) all the roots of the equation $f(x)=0$ are real.
1991 Irish Math Olympiad
Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$.



(a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$.

(b) Find an example of eleven consecutive squares whose sum is a square.



Can anyone help me with this?

Thanks.
1991 Irish Math Olympiad
Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, \quad n=1,2,3,\dots$$and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$. Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$for all positive integers $n$.
1991 Irish Math Olympiad
Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG = DE$. Prove that $ CA = CB$.



Original formulation:



Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF = DE,$ prove that $ AC = BC.$
1991 Irish Math Olympiad
Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$and $$f(2x)=2f(f(x))$$for all $x\in\mathbb{P}$.

Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.
1991 Irish Math Olympiad
Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties:



(a) $0$ is not in $S$;



(b) for each $s_1,s_2$ in $S$, the rational number $s_1/s_2$ is in $S$;



(c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$.



Prove that if $x$ belongs to $S$, then there exists elements $y,z$ in $S$ such that $x=y+z$.
1992 Irish Math Olympiad
Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.
1992 Irish Math Olympiad
How many ordered triples $(x,y,z)$ of real numbers satisfy the system of equations $$x^2+y^2+z^2=9,$$$$x^4+y^4+z^4=33,$$$$xyz=-4?$$
1992 Irish Math Olympiad
In a triangle $ABC$, the points $A’$, $B’$ and $C’$ on the sides opposite $A$ ,$B$ and $C$, respectively, are such that the lines $AA’$, $BB’$ and $CC’$ are concurrent. Prove that the diameter of the circumscribed circle of the triangle $ABC$ equals the product $|AB’|\cdot |BC’|\cdot |CA’|$ divided by the area of the triangle $A’B’C’$.
1992 Irish Math Olympiad
Let $ABC$ be a triangle such that the coordinates of the points $A$ and $B$ are rational numbers. Prove that the coordinates of $C$ are rational if, and only if, $\tan A$, $\tan B$, and $\tan C$, when defined, are all rational numbers.
1992 Irish Math Olympiad
Let $n > 2$ be an integer and let $m = \sum k^3$, where the sum is taken over all integers $k$ with $1 \leq k < n$ that are relatively prime to $n$. Prove that $n$ divides $m$.
1992 Irish Math Olympiad
If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a digital root of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.
1992 Irish Math Olympiad
Let $a, b, c$ and $d$ be real numbers with $a \neq 0$. Prove that if all the roots of the cubic equation



$az^{3} +bz^{2} +cz+d=0$



lie to the left of the imaginary axis in the complex plane, then



$ab >0, bc-ad >0, ad>0$.
1992 Irish Math Olympiad
A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.
1992 Irish Math Olympiad
If, for $k=1,2,\dots ,n$, $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$
1993 Irish Math Olympiad
The real numbers $\alpha ,\beta$ satisfy the equations $$\alpha^3-3\alpha^2+5\alpha-17=0$$$$\beta^3-3\beta^2+5\beta+11=0$$Find $\alpha+\beta$.
1993 Irish Math Olympiad
A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1+a_2+...+a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10=5+2+1+1+1=5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.
1993 Irish Math Olympiad
A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.
1993 Irish Math Olympiad
Let $ f(x)=x^n+a_{n-1} x^{n-1}+...+a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|=f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.
1993 Irish Math Olympiad
For a complex number $ z=x+iy$ we denote by $ P(z)$ the corresponding point $ (x,y)$ in the plane. Suppose $ z_1,z_2,z_3,z_4,z_5,\alpha$ are nonzero complex numbers such that:

$ (i)$ $ P(z_1),...,P(z_5)$ are vertices of a complex pentagon $ Q$ containing the origin $ O$ in its interior, and

$ (ii)$ $ P(\alpha z_1),...,P(\alpha z_5)$ are all inside $ Q$.

If $ \alpha=p+iq$ $ (p,q \in \mathbb{R})$, prove that $ p^2+q^2 \le 1$ and $ p+q \tan \frac{\pi}{5} \le 1$.