Navvye/Olympiad-Problems-CSV huggingface dataset:
1993 Irish Math Olympiad
Show that among any five points $ P_1,...,P_5$ with integer coordinates in the plane, there exists at least one pair $ (P_i,P_j)$, with $ i \not= j$ such that the segment $ P_i P_j$ contains a point $ Q$ with integer coordinates other than $ P_i, P_j$.
1993 Irish Math Olympiad
Let $ a_i,b_i$ $ (i=1,2,...,n)$ be real numbers such that the $ a_i$ are distinct, and suppose that there is a real number $ \alpha$ such that the product $ (a_i+b_1)(a_i+b_2)...(a_i+b_n)$ is equal to $ \alpha$ for each $ i$. Prove that there is a real number $ \beta$ such that $ (a_1+b_j)(a_2+b_j)...(a_n+b_j)$ is equal to $ \beta$ for each $ j$.
1993 Irish Math Olympiad
If $ 1 \le r \le n$ are integers, prove the identity:



$ \displaystyle\sum_{d=1}^{\infty}\binom {n-r+1}{d} \binom {r-1} {d-1}=\binom {n}{r}.$
1993 Irish Math Olympiad
Let $x$ be a real number with $0<x<\pi $.Prove that, for all natural number $n$ ,$$sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.$$
1993 Irish Math Olympiad
$ (a)$ The rectangle $ PQRS$ with $ PQ=l$ and $ QR=m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l+m-d$ of these squares, where $ d=(l,m)$.

$ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?
1994 Irish Math Olympiad
Let $ x,y$ be positive integers with $ y>3$ and $ x^2+y^4=2((x-6)^2+(y+1)^2).$ Prove that: $ x^2+y^4=1994.$
1994 Irish Math Olympiad
Let $ A,B,C$ be collinear points on the plane with $ B$ between $ A$ and $ C$. Equilateral triangles $ ABD,BCE,CAF$ are constructed with $ D,E$ on one side of the line $ AC$ and $ F$ on the other side. Prove that the centroids of the triangles are the vertices of an equilateral triangle, and that the centroid of this triangle lies on the line $ AC$.
1994 Irish Math Olympiad
Find all real polynomials $ f(x)$ satisfying $ f(x^2)=f(x)f(x-1)$ for all $ x$.
1994 Irish Math Olympiad
Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.
1994 Irish Math Olympiad
Let $ f(n)$ be defined for $ n \in \mathbb{N}$ by $ f(1)=2$ and $ f(n+1)=f(n)^2-f(n)+1$ for $ n \ge 1$. Prove that for all $ n >1:$



$ 1-\frac{1}{2^{2^{n-1}}}<\frac{1}{f(1)}+\frac{1}{f(2)}+...+\frac{1}{f(n)}<1-\frac{1}{2^{2^n}}$
1994 Irish Math Olympiad
A sequence $ (x_n)$ is given by $ x_1=2$ and $ nx_n=2(2n-1)x_{n-1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.
1994 Irish Math Olympiad
Let $ p,q,r$ be distinct real numbers that satisfy: $ q=p(4-p), \, r=q(4-q), \, p=r(4-r).$ Find all possible values of $ p+q+r$.
1994 Irish Math Olympiad
Prove that for every integer $ n>1$,



$ n((n+1)^{\frac{2}{n}}-1)<\displaystyle\sum_{i=1}^{n}\frac{2i+1}{i^2}<n(1-n^{-\frac{2}{n-1}})+4$.
1994 Irish Math Olympiad
Suppose that $ \omega, a,b,c$ are distinct real numbers for which there exist real numbers $ x,y,z$ that satisfy the following equations:



$ x+y+z=1,$

$ a^2 x+b^2 y +c^2 z=\omega ^2,$

$ a^3 x+b^3 y +c^3 z=\omega ^3,$

$ a^4 x+b^4 y +c^4 z=\omega ^4.$



Express $ \omega$ in terms of $ a,b,c$.
1994 Irish Math Olympiad
If a square is partitioned into $ n$ convex polygons, determine the maximum possible number of edges in the obtained figure.



(You may wish to use the following theorem of Euler: If a polygon is partitioned into $ n$ polygons with $ v$ vertices and $ e$ edges in the resulting figure, then $ v-e+n=1$.)
1995 Irish Math Olympiad
There are $ n^2$ students in a class. Each week all the students participate in a table quiz. Their teacher arranges them into $ n$ teams of $ n$ players each. For as many weeks as possible, this arrangement is done in such a way that any pair of students who were members of the same team one week are not in the same team in subsequent weeks. Prove that after at most $ n+2$ weeks, it is necessary for some pair of students to have been members of the same team in at least two different weeks.
1995 Irish Math Olympiad
Determine all integers $ a$ for which the equation $ x^2+axy+y^2=1$ has infinitely many distinct integer solutions $ x,y$.
1995 Irish Math Olympiad
Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality:



$ 2AD \ge \sqrt{3} (AB+BC+CD)$.
1995 Irish Math Olympiad
Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point $ j$ at which at least two disks are piled and then takes two disks from the point $ j$ and places one of them at $ j-1$ and one at $ j+1$. Initially, $ 2n+1$ disks are placed at point $ 0$. The player proceeds to perform moves as long as possible. Prove that after $ \frac{1}{6} n(n+1)(2n+1)$ moves no further moves will be possible and that at this stage, one disks remains at each of the positions $ -n,-n+1,...,0,...n$.
1995 Irish Math Olympiad
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:



$ x f(x)-yf(y)=(x-y)f(x+y)$.
1995 Irish Math Olympiad
Prove that for every positive integer $ n$,



$ n^n \le (n!)^2 \le \left( \frac{(n+1)(n+2)}{6} \right) ^n.$
1995 Irish Math Olympiad
Let $ a,b,c$ be complex numbers. Prove that if all the roots of the equation $ x^3+ax^2+bx+c=0$ are of module $ 1$, then so are the roots of the equation $ x^3+|a|x^2+|b|x+|c|=0$.
1995 Irish Math Olympiad
Let $S$ be the square consisting of all pints $(x,y)$ in the plane with $0\le x,y\le 1$. For each real number $t$ with $0<t<1$, let $C_t$ denote the set of all points $(x,y)\in S$ such that $(x,y)$ is on or above the line joining $(t,0)$ to $(0,1-t)$.

Prove that the points common to all $C_t$ are those points in $S$ that are on or above the curve $\sqrt{x}+\sqrt{y}=1$.
1995 Irish Math Olympiad
Points $ P,Q,R$ are given in the plane. It is known that there is a triangle $ ABC$ such that $ P$ is the midpoint of $ BC$, $ Q$ the point on side $ CA$ with $ \frac{CQ}{QA}=2$, and $ R$ the point on side $ AB$ with $ \frac{AR}{RB}=2$. Determine with proof how the triangle $ ABC$ may be reconstructed from $ P,Q,R$.
1995 Irish Math Olympiad
For each integer $ n$ of the form $ n=p_1 p_2 p_3 p_4$, where $ p_1,p_2,p_3,p_4$ are distinct primes, let $ 1=d_1<d_2<...<d_{15}<d_{16}=n$ be the divisors of $ n$. Prove that if $ n<1995$, then $ d_9-d_8 \not= 22$.
1996 Irish Math Olympiad
For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!+1$ and $ (n+1)!$. Find, without proof, a formula for $ f(n)$.
1996 Irish Math Olympiad
Let $ S(n)$ denote the sum of the digits of a natural number $ n$ (in base $ 10$). Prove that for every $ n$, $ S(2n) \le 2S(n) \le 10S(2n)$. Prove also that there is a positive integer $ n$ with $ S(n)=1996S(3n)$.
1996 Irish Math Olympiad
A function $ f$ from $ [0,1]$ to $ \mathbb{R}$ has the following properties:

$ (i)$ $ f(1)=1;$

$ (ii)$ $ f(x) \ge 0$ for all $ x \in [0,1]$;

$ (iii)$ If $ x,y,x+y \in [0,1]$, then $ f(x+y) \ge f(x)+f(y)$.

Prove that $ f(x) \le 2x$ for all $ x \in [0,1]$.
1996 Irish Math Olympiad
Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.
1996 Irish Math Olympiad
Show how to dissect a square into at most five pieces in such a way that the pieces can be reassembled to form three squares of (pairwise) distinct areas.
1996 Irish Math Olympiad
The Fibonacci sequence is defined by $ F_0=0, F_1=1$ and $ F_{n+2}=F_n+F_{n+1}$ for $ n \ge 0$. Prove that:

$ (a)$ The statement $ "F_{n+k}-F_n$ is divisible by $ 10$ for all $ n \in \mathbb{N}"$ is true if $ k=60$ but false for any positive integer $ k<60$.

$ (b)$ The statement $ "F_{n+t}-F_n$ is divisible by $ 100$ for all $ n \in \mathbb{N}"$ is true if $ t=300$ but false for any positive integer $ t<300$.
1996 Irish Math Olympiad
Show that for every positive integer $ n$, $ 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4$.
1996 Irish Math Olympiad
Suppose that $ p$ is a prime number and $ a$ and $ n$ positive integers such that: $ 2^p+3^p=a^n$. Prove that $ n=1$.
1996 Irish Math Olympiad
In an acute-angled triangle $ ABC$, $ D,E,F$ are the feet of the altitudes from $ A,B,C$, respectively, and $ P,Q,R$ are the feet of the perpendiculars from $ A,B,C$ onto $ EF,FD,DE$, respectively. Prove that the lines $ AP,BQ,CR$ are concurrent.
1996 Irish Math Olympiad
The following game is played on a rectangular chessboard $ 5 \times 9$ (with five rows and nine columns). Initially, a number of discs are randomly placed on some of the squares of the chessboard, with at most one disc on each square. A complete move consists of the moving every disc subject to the following rules:

$ (1)$ Each disc may be moved one square up, down, left or right;

$ (2)$ If a particular disc is moved up or down as part of a complete move, then it must be moved left or right in the next complete move;

$ (3)$ If a particular disc is moved left or right as part of a complete move, then it must be moved up or down in the next complete move;

$ (4)$ At the end of a complete move, no two discs can be on the same square.

The game stops if it becomes impossible to perform a complete move. Prove that if initially $ 33$ discs are placed on the board then the game must eventually stop. Prove also that it is possible to place $ 32$ discs on the boards in such a way that the game could go on forever.
1997 Irish Math Olympiad
Find all pairs of integers $ (x,y)$ satisfying $ 1+1996x+1998y=xy.$
1997 Irish Math Olympiad
For a point $ M$ inside an equilateral triangle $ ABC$, let $ D,E,F$ be the feet of the perpendiculars from $ M$ onto $ BC,CA,AB$, respectively. Find the locus of all such points $ M$ for which $ \angle FDE$ is a right angle.
1997 Irish Math Olympiad
Find all polynomials $ p(x)$ satisfying the equation: $ (x-16)p(2x)=16(x-1)p(x)$ for all $ x$.
1997 Irish Math Olympiad
Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a+b+c\ge abc$. Prove that:



$ a^2+b^2+c^2 \ge abc.$
1997 Irish Math Olympiad
Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) +1}$. For $ a,b \in S$, define:



$ a \ast b=2^{\delta (a)-1} (b-3)+a.$



Prove that if $ a,b,c \in S$, then:

$ (a)$ $ a \ast b \in S$ and

$ (b)$ $ (a \ast b)\ast c=a \ast (b \ast c)$.
1997 Irish Math Olympiad
Given a positive integer $ n$, denote by $ \sigma (n)$ the sum of all positive divisors of $ n$. We say that $ n$ is $ abundant$ if $ \sigma (n)>2n.$ (For example, $ 12$ is abundant since $ \sigma (12)=28>2 \cdot 12$.) Let $ a,b$ be positive integers and suppose that $ a$ is abundant. Prove that $ ab$ is abundant.
1997 Irish Math Olympiad
A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$. If $ \angle A=\angle B=120^{\circ}, \angle D=90^{\circ}$ and $ BC=1$, find, with proof, the length of $ AD$.
1997 Irish Math Olympiad
Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k=0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a+b$ is a power of $ 2$.
1997 Irish Math Olympiad
Let $ S$ be the set of natural numbers $ n$ satisfying the following conditions:



$ (i)$ $ n$ has $ 1000$ digits,

$ (ii)$ all the digits of $ n$ are odd, and

$ (iii)$ any two adjacent digits of $ n$ differ by $ 2$.



Determine the number of elements of $ S$.
1997 Irish Math Olympiad
Let $ p$ be an odd prime number and $ n$ a natural number. Then $ n$ is called $ p-partitionable$ if $ T=\{1,2,...,n \}$ can be partitioned into (disjoint) subsets $ T_1,T_2,...,T_p$ with equal sums of elements. For example, $ 6$ is $ 3$-partitionable since we can take $ T_1=\{ 1,6 \}$, $ T_2=\{ 2,5 \}$ and $ T_3=\{ 3,4 \}$.

$ (a)$ Suppose that $ n$ is $ p$-partitionable. Prove that $ p$ divides $ n$ or $ n+1$.

$ (b)$ Suppose that $ n$ is divisible by $ 2p$. Prove that $ n$ is $ p$-partitionable.
1998 Irish Math Olympiad
Prove that if $ x \not= 0$ is a real number, then: $ x^8-x^5-\frac{1}{x}+\frac{1}{x^4} \ge 0$.
1998 Irish Math Olympiad
The distances from a point $ P$ inside an equilateral triangle to the vertices of the triangle are $ 3,4$, and $ 5$. Find the area of the triangle.
1998 Irish Math Olympiad
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$.
1998 Irish Math Olympiad
Show that a disk of radius $ 2$ can be covered by seven (possibly overlapping) disks of radius $ 1$.
1998 Irish Math Olympiad
If $ x$ is a real number such that $ x^2-x$ and $ x^n-x$ are integers for some $ n \ge 3$, prove that $ x$ is an integer.
1998 Irish Math Olympiad
Find all positive integers $ n$ having exactly $ 16$ divisors $ 1=d_1<d_2<...<d_{16}=n$ such that $ d_6=18$ and $ d_9-d_8=17.$
1998 Irish Math Olympiad
Prove that if $ a,b,c$ are positive real numbers, then:



$ \frac{9}{a+b+c} \le 2 \left( \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \right) \le \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$
1998 Irish Math Olympiad
$ (a)$ Prove that $ \mathbb{N}$ can be partitioned into three (mutually disjoint) sets such that, if $ m,n \in \mathbb{N}$ and $ |m-n|$ is $ 2$ or $ 5$, then $ m$ and $ n$ are in different sets.

$ (b)$ Prove that $ \mathbb{N}$ can be partitioned into four sets such that, if $ m,n \in \mathbb{N}$ and $ |m-n|$ is $ 2,3,$ or $ 5$, then $ m$ and $ n$ are in different sets. Show, however, that $ \mathbb{N}$ cannot be partitioned into three sets with this property.
1998 Irish Math Olympiad
A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n+2}=\frac{1+x_{n+1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.
1998 Irish Math Olympiad
A triangle $ ABC$ has integer sides, $ \angle A=2 \angle B$ and $ \angle C>90^{\circ}$. Find the minimum possible perimeter of this triangle.
1999 Irish Math Olympiad
Find all real numbers $ x$ which satisfy: $ \frac{x^2}{(x+1-\sqrt{x+1})^2}<\frac{x^2+3x+18}{(x+1)^2}.$
1999 Irish Math Olympiad
Show that there is a positive number in the Fibonacci sequence which is divisible by $ 1000$.
1999 Irish Math Olympiad
If $ AD$ is the altitude, $ BE$ the angle bisector, and $ CF$ the median of a triangle $ ABC$, prove that $ AD,BE,$ and $ CF$ are concurrent if and only if:



$ a^2(a-c)=(b^2-c^2)(a+c),$



where $ a,b,c$ are the lengths of the sides $ BC,CA,AB$, respectively.
1999 Irish Math Olympiad
A $ 100 \times 100$ square floor consisting of $ 10000$ squares is to be tiled by rectangular $ 1 \times 3$ tiles, fitting exactly over three squares of the floor.

$ (a)$ If a $ 2 \times 2$ square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles.

$ (b)$ If, instead, a $ 2 \times 2$ square is removed from the corner, prove that such a tiling is not possble.
1999 Irish Math Olympiad
The sequence $ u_n$, $ n=0,1,2,...$ is defined by $ u_0=0, u_1=1$ and for each $ n \ge 1$, $ u_{n+1}$ is the smallest positive integer greater than $ u_n$ such that $ \{ u_0,u_1,...,u_{n+1} \}$ contains no three elements in arithmetic progression. Find $ u_{100}$.
1999 Irish Math Olympiad
Solve the system of equations:



$ y^2=(x+8)(x^2+2),$

$ y^2-(8+4x)y+(16+16x-5x^2)=0.$
1999 Irish Math Olympiad
A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies:

$ (a)$ $ f(ab)=f(a)f(b)$ whenever $ a$ and $ b$ are coprime;

$ (b)$ $ f(p+q)=f(p)+f(q)$ for all prime numbers $ p$ and $ q$.

Prove that $ f(2)=2,f(3)=3$ and $ f(1999)=1999.$
1999 Irish Math Olympiad
The sum of positive real numbers $ a,b,c,d$ is $ 1$. Prove that:



$ \frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2},$



with equality if and only if $ a=b=c=d=\frac{1}{4}$.
1999 Irish Math Olympiad
Find all positive integers $ m$ with the property that the fourth power of the number of (positive) divisors of $ m$ equals $ m$.
1999 Irish Math Olympiad
A convex hexagon $ ABCDEF$ satisfies $ AB=BC, CD=DE, EF=FA$ and: $ \angle ABC+\angle CDE+\angle EFA = 360^{\circ}$. Prove that the perpendiculars from $ A,C$ and $ E$ to $ FB,BD$ and $ DF$ respectively are concurrent.
2000 Irish Math Olympiad
Consider the set $ S$ of all numbers of the form $ a(n)=n^2+n+1, n \in \mathbb{N}.$ Show that the product $ a(n)a(n+1)$ is in $ S$ for all $ n \in \mathbb{N}$ and give an example of two elements $ s,t$ of $ S$ such that $ s,t \notin S$.
2000 Irish Math Olympiad
Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD = \angle HFD = 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that:



$ FG = t = \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a = \frac {t \sqrt {3}}{2 + \sqrt {3}}$.
2000 Irish Math Olympiad
Let $ f(x)=5x^{13}+13x^5+9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$.
2000 Irish Math Olympiad
The sequence $ a_1<a_2<...<a_M$ of real numbers is called a weak arithmetic progression of length $ M$ if there exists an arithmetic progression $ x_0,x_1,...,x_M$ such that:



$ x_0 \le a_1<x_1 \le a_2<x_2 \le ... \le a_M<x_M.$



$ (a)$ Prove that if $ a_1<a_2<a_3$ then $ (a_1,a_2,a_3)$ is a weak arithmetic progression.

$ (b)$ Prove that any subset of $ \{ 0,1,2,...,999 \}$ with at least $ 730$ elements contains a weak arithmetic progression of length $ 10$.
2000 Irish Math Olympiad
Consider all parabolas of the form $ y=x^2+2px+q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.
2000 Irish Math Olympiad
Prove that if $ x,y$ are nonnegative real numbers with $ x+y=2$, then: $ x^2 y^2 (x^2+y^2) \le 2$.
2000 Irish Math Olympiad
In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that:



$ R^2=\frac{(ab+cd)(ac+bd)(ad+bc)}{16Q^2}$.



Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.
2000 Irish Math Olympiad
For each positive integer $ n$ find all positive integers $ m$ for which there exist positive integers $ x_1<x_2<...<x_n$ with:



$ \frac{1}{x_1}+\frac{2}{x_2}+...+\frac{n}{x_n}=m.$
2000 Irish Math Olympiad
Show that in each set of ten consecutive integers there is one that is coprime with each of the other integers. (For example, in the set $ \{ 114,115,...,123 \}$ there are two such numbers: $ 119$ and $ 121.)$
2000 Irish Math Olympiad
Let $ p(x)=a_0 +a_1 x+...+a_n x^n$ be a polynomial with nonnegative real coefficients. Suppose that $ p(4)=2$ and $ p(16)=8$. Prove that $ p(8) \le 4$ and find all such $ p$ with $ p(8)=4$.
2001 Irish Math Olympiad
Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n=a!+b!+c!$.
2001 Irish Math Olympiad
Let $ ABC$ be a triangle with sides $ BC=a, CA=b,AB=c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2+c^2=5a^2$.
2001 Irish Math Olympiad
Show that if an odd prime number $ p$ can be expressed in the form $ x^5-y^5$ for some integers $ x,y,$ then:



$ \sqrt{\frac{4p+1}{5}}=\frac{v^2+1}{2}$ for some odd integer $ v$.
2001 Irish Math Olympiad
Prove that for all positive integers $ n$: $ \frac{2n}{3n+1} \le \displaystyle\sum_{k=n+1}^{2n}\frac{1}{k} \le \frac{3n+1}{4(n+1)}$.
2001 Irish Math Olympiad
Prove that for all real numbers $ a,b$ with $ ab>0$ we have:



$ \sqrt[3]{\frac{a^2 b^2 (a+b)^2}{4}} \le \frac{a^2+10ab+b^2}{12}$



and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$



$ \sqrt[3]{\frac{a^2 b^2 (a+b)^2}{4}} \le \frac{a^2+ab+b^2}{3}$



and find the cases of equality.
2001 Irish Math Olympiad
Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n+a \cdot 32^n$ for some odd $ n$.
2001 Irish Math Olympiad
Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with $ 20$ beads, $ 10$ of which are black and $ 10$ are white. On each hoop the positions of the beads are labelled $ 1$ through $ 20$ as shown. We say there is a match at position $ i$ if all three beads at position $ i$ have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than $ 5$ matches.
2001 Irish Math Olympiad
In an acute-angled triangle $ ABC$, $ D$ is the foot of the altitude from $ A$, and $ P$ a point on segment $ AD$. The lines $ BP$ and $ CP$ meet $ AC$ and $ AB$ at $ E$ and $ F$ respectively. Prove that $ AD$ bisects the angle $ EDF$.
2001 Irish Math Olympiad
Find all nonnegative real numbers $ x$ for which $ \sqrt[3]{13+\sqrt{x}}+\sqrt[3]{13-\sqrt{x}}$ is an integer.
2001 Irish Math Olympiad
Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy:



$ f(x+f(y))=f(x)+y$ for all $ x,y \in \mathbb{N}$.
2002 Irish Math Olympiad
In a triangle $ ABC$ with $ AB=20, AC=21$ and $ BC=29$, points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD=8$ and $ EC=9$. Calculate the angle $ \angle DAE$.
2002 Irish Math Olympiad
$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?

$ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?
2002 Irish Math Olympiad
Find all triples of positive integers $ (p,q,n)$, with $ p$ and $ q$ primes, satisfying:



$ p(p+3)+q(q+3)=n(n+3)$.
2002 Irish Math Olympiad
The sequence $ (a_n)$ is defined by $ a_1=a_2=a_3=1$ and $ a_{n+1}a_{n-2}-a_n a_{n-1}=2$ for all $ n \ge 3.$ Prove that $ a_n$ is a positive integer for all $ n \ge 1$.
2002 Irish Math Olympiad
Let $ 0<a,b,c<1$. Prove the inequality:



$ \frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c} \ge \frac {3 \sqrt[3]{abc}}{1- \sqrt[3]{abc}}.$



Determine the cases of equality.
2002 Irish Math Olympiad
A $ 3 \times n$ grid is filled as follows. The first row consists of the numbers from $ 1$ to $ n$ arranged in ascending order. The second row is a cyclic shift of the top row: $ i,i+1,...,n,1,2,...,i-1$ for some $ i$. The third row has the numbers $ 1$ to $ n$ in some order so that in each of the $ n$ columns, the sum of the three numbers is the same. For which values of $ n$ is it possible to fill the grid in this way? For all such $ n$, determine the number of different ways of filling the grid.
2002 Irish Math Olympiad
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:

$ (i)$ $ a+c=d;$

$ (ii)$ $ a(a+b+c+d)=c(d-b);$

$ (iii)$ $ 1+bc+d=bd$.

Determine $ n$.
2002 Irish Math Olympiad
Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that:



$ f(x+f(y))=y+f(x)$ for all $ x,y \in \mathbb{Q}$.
2002 Irish Math Olympiad
Let $ \alpha=2+\sqrt{3}$. Prove that $ \alpha^n-[\alpha^n]=1-\alpha^{-n}$ for all $ n \in \mathbb{N}_0$.
2002 Irish Math Olympiad
Let $ ABC$ be a triangle with integer side lengths, and let its incircle touch $ BC$ at $ D$ and $ AC$ at $ E$. If $ |AD^2-BE^2| \le 2$, show that $ AC=BC$.
2003 Irish Math Olympiad
find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$