Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Irish Math Olympiad
Let $$C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$$and let

$$S=\{4,5,9,14,23,37\}$$Find two sets $A$ and $B$ with the properties

(a) $A\cap B=\emptyset$.

(b) $A\cup B=C$.

(c) The sum of two distinct elements of $A$ is not in $S$.

(d) The sum of two distinct elements of $B$ is not in $S$.
2012 Irish Math Olympiad
$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$. $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$. $X$ is a point on the circumference of the circle between $A$ and $D$. $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$. Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$.
2012 Irish Math Olympiad
Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.
2012 Irish Math Olympiad
There exists an infinite set of triangles with the following properties:



(a) the lengths of the sides are integers with no common factors, and



(b) one and only one angle is $60^\circ$.



One such triangle has side lengths $5,7,8$. Find two more.
2012 Irish Math Olympiad
(a) Show that if $x$ and $y$ are positive real numbers, then $$(x+y)^5\ge 12xy(x^3+y^3)$$

(b) Prove that the constant $12$ is the best possible. In other words, prove that for any $K>12$ there exist positive real numbers $x$ and $y$ such that $$(x+y)^5<Kxy(x^3+y^3)$$
2012 Irish Math Olympiad
Let $S(n)$ be the sum of the decimal digits of $n$. For example. $S(2012)=2+0+1+2=5$. Prove that there is no integer $n>0$ for which $n-S(n)=9990$.
2012 Irish Math Olympiad
Consider a triangle $ABC$ with $|AB|\neq |AC|$. The angle bisector of the angle $CAB$ intersects the circumcircle of $\triangle ABC$ at two points $A$ and $D$. The circle of center $D$ and radius $|DC|$ intersects the line $AC$ at two points $C$ and $B’$. The line $BB’$ intersects the circumcircle of $\triangle ABC$ at $B$ and $E$. Prove that $B’$ is the orthocenter of $\triangle AED$.
2012 Irish Math Olympiad
Suppose $a,b,c$ are positive numbers. Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2\ge (2a+b+c) \left(\frac{2}{a}+\frac{1}{b}+\frac{1}{c}\right)$$with equality if and only if $a=b=c$.
2012 Irish Math Olympiad
Let $x$ > $1$ be an integer. Prove that $x^5$ + $x$ + $1$ is divisible by at least two distinct prime numbers.
2012 Irish Math Olympiad
Let $n$ be a positive integer. A mouse sits at each corner point of an $n\times n$ board, which is divided into unit squares as shown below for the example $n=5$.



[asy]

unitsize(5mm);

defaultpen(linewidth(.5pt));

fontsize(25pt);

for(int i=0; i<=5; ++i)

{

for(int j=0; j<=5; ++j)

{

draw((0,i)--(5,i));

draw((j,0)--(j,5));

}}

dot((0,0));

dot((5,0));

dot((0,5));

dot((5,5));





[/asy]



The mice then move according to a sequence of steps, in the following manner:



(a) In each step, each of the four mice travels a distance of one unit in a horizontal or vertical direction. Each unit distance is called an edge of the board, and we say that each mouse uses an edge of the board.



(b) An edge of the board may not be used twice in the same direction.



(c) At most two mice may occupy the same point on the board at any time.



The mice wish to collectively organize their movements so that each edge of the board will be used twice (not necessarily be the same mouse), and each mouse will finish up at its starting point. Determine, with proof, the values of $n$ for which the mice may achieve this goal.
2013 Irish Math Olympiad
Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.
2013 Irish Math Olympiad
Prove that $$ 1-\frac{1}{2012}\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2013}\right)\ge \frac{1}{\sqrt[2012]{2013}}.$$
2013 Irish Math Olympiad
The altitudes of a triangle  $\triangle ABC$ are used to form the sides of a second triangle  $\triangle A_1B_1C_1$. The altitudes of $\triangle A_1B_1C_1$ are then used to form the sides of a third triangle  $\triangle A_2B_2C_2$. Prove that  $\triangle A_2B_2C_2$ is similar to  $\triangle ABC$.
2013 Irish Math Olympiad
Each of the $36$ squares of a $6 \times 6$ table is to be coloured either Red, Yellow or Blue.

(a) No row or column is contain more than two squares of the same colour.

(b) In any four squares obtained by intersecting two rows with two columns, no colour is to occur exactly three times.

In how many dierent ways can the table be coloured if both of these rules are to be respected?
2013 Irish Math Olympiad
$A, B$ and $C$ are points on the circumference of a circle with centre $O$. Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$. The two tangents intersect at $K$. The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.
2013 Irish Math Olympiad
The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on

the line BD is such that $|AB| = |AC| = |CD|.$

Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$

.
2013 Irish Math Olympiad
Consider the collection of different squares which may be formed by sets of four points chosen from the $12$ labelled

points in the diagram on the right.

For each possible area such a square may have, determine the number of squares which have this area.

Make sure to explain why your list is complete.
2013 Irish Math Olympiad
Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$.
2013 Irish Math Olympiad
We say that a doubly infinite sequence

$. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$

is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n.

(a) Find a subaveraging sequence in which all entries are different from each other. Prove that all

entries are indeed distinct.

(b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such

that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .
2013 Irish Math Olympiad
Let $a,b,c $ be  real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$

and $S=2x^3-9xy+9z .$

$(a)$  Prove that  $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t $ , respectively , for any  real number $t$.

$(b)$ Prove that  $ (3y-x^2)^3\ge S^2 .$
2014 Irish Math Olympiad
Given an $8\times 8$ chess board, in how many ways can we select $56$ squares on the board while satisfying both of the following requirements:

(a) All black squares are selected.

(b) Exactly seven squares are selected in each column and in each row.
2014 Irish Math Olympiad




Quote:

Prove that for $N>1$ that $(N^{2})^{2014}  - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$



reason for this commentcommenting only to copy the formulation for the posts collection, hide comment shall be removed tomorrow
2014 Irish Math Olympiad
In the triangle ABC, D is the foot of the altitude from A to BC, and M is the midpoint of the line

segment BC. The three angles ∠BAD, ∠DAM and ∠MAC are all equal. Find the angles of the

triangle ABC.
2014 Irish Math Olympiad
Three different non-zero real numbers $a,b,c$ satisfy the equations $a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}=p $, where $p$ is a real number. Prove that  $abc+2p=0.$
2014 Irish Math Olympiad
Suppose $a_1,a_2,\ldots,a_n>0 $, where $n>1$ and $\sum_{i=1}^{n}a_i=1$.

For each $i=1,2,\ldots,n $, let $b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}$. Prove that $$\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .$$

When does equality occur ?
2014 Irish Math Olympiad
Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly$ 20$ dierent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$. Find all possible values of $N$.(Both $1$ and $m$ are counted as divisors of the number $m$.)
2014 Irish Math Olympiad
The square $ABCD$ is inscribed in a circle with center $O$. Let $E$ be the midpoint of $AD$. The line $CE$ meets the circle again at $F$. The lines $FB$ and $AD$ meet at $H$. Prove $HD = 2AH$
2014 Irish Math Olympiad
(a) Let $a_0, a_1,a_2$ be real numbers and consider the polynomial $P(x) = a_0 + a_1x + a_2x^2$ .

Assume that $P(-1), P(0)$ and $P(1)$ are integers.

Prove that $P(n)$ is an integer for all integers $n$.

(b) Let $a_0,a_1, a_2, a_3$ be real numbers and consider the polynomial $Q(x) = a0 + a_1x + a_2x^2 + a_3x^3 $.

Assume that there exists an integer $i$ such that $Q(i),Q(i+1),Q(i+2)$ and $Q(i+3)$ are integers.

Prove that $Q(n)$ is an integer for all integers $n$.
2014 Irish Math Olympiad
Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers.

Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ .

Let $a_0$ be a real number and let

$f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ .

Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$.
2014 Irish Math Olympiad
Over a period of $k$ consecutive days, a total of $2014$ babies were born in a certain city, with at least one baby being born each day. Show that:

(a) If $1014 < k \le 2014$, there must be a period of consecutive days during which exactly $100$ babies were born.

(b) By contrast, if $k = 1014$, such a period might not exist.
2015 Irish Math Olympiad
In the triangle $ABC$, the length of the altitude from $A$ to $BC$ is equal to $1$. $D$ is the midpoint of $AC$. What are the possible lengths of $BD$?
2015 Irish Math Olympiad
A regular polygon with $n \ge 3$ sides is given. Each vertex is coloured either red, green or blue, and no two adjacent vertices of the polygon are the same colour. There is at least one vertex of each colour.

Prove that it is possible to draw certain diagonals of the polygon in such a way that they intersect only at the vertices of the polygon and they divide the polygon into triangles so that each such triangle has vertices of three different colours.
2015 Irish Math Olympiad
Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.
2015 Irish Math Olympiad
Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that

(a) $|LH| = |KM|$

(b) the line through $B$ perpendicular to $DE$ bisects $HK$.
2015 Irish Math Olympiad
Suppose a doubly infinite sequence of real numbers $. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .$ has the property that $$a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},$$for all integers $n .$ Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \leq R$ for all $n$), then $a_n$ has the same value for all $n.$
2015 Irish Math Olympiad
Suppose $x,y$ are nonnegative real numbers such that $x + y \le 1$. Prove that $8xy \le  5x(1 - x) + 5y(1 - y)$

and determine the cases of equality.
2015 Irish Math Olympiad
Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$.

A nonempty subset $S$ of  $\Omega$  is called sum-free if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition.

Prove that $\Omega$  has more than $2^n$ distinct sum-free subsets.
2015 Irish Math Olympiad
In triangle $\triangle ABC$, the angle $\angle BAC$ is less than $90^o$. The perpendiculars from $C$ on $AB$ and from $B$ on $AC$ intersect the circumcircle of $\triangle ABC$ again at $D$ and $E$ respectively. If $|DE| =|BC|$, find the measure of the angle $\angle BAC$.
2015 Irish Math Olympiad
Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.
2015 Irish Math Olympiad
Prove that, for all pairs of nonnegative integers, $j,n$, $$\sum_{K=0}^{n}k^j\binom n k \ge 2^{n-j} n^j$$
2016 Irish Math Olympiad
If the three-digit number $ABC$ is divisible by $27$, prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$.
2016 Irish Math Olympiad
In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle  ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle  BAC$.
2016 Irish Math Olympiad
Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?
2016 Irish Math Olympiad
Let $ABC$ be a triangle with $|AC| \ne |BC|$. Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$, so that $P$ is between $A$ and $B$. Prove that if $M$ is any point on the circle with diameter $PQ$, then $\angle AMP = \angle BMP$.
2016 Irish Math Olympiad
Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$, such that $a_1 + a_2 + ...+ a_m = 10m$. Prove that $(a_1a_2a_3 \cdot ...\cdot  a_m)^{1/m} \le 3\sqrt{11}$.
2016 Irish Math Olympiad
Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.
2016 Irish Math Olympiad
A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$. Within each row, all entries are distinct. What is the maximum possible number of columns?
2016 Irish Math Olympiad
Suppose $a, b, c$ are real numbers such that $abc \ne 0$.

Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$.

Prove also that    $\frac{1 - x^2}{a^2} =  \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.
2016 Irish Math Olympiad
Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$

is an integer and find its value
2016 Irish Math Olympiad
Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$.



Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.
2017 Irish Math Olympiad
Problem 1 (Start of paper 1, time for paper : 3 hours)

1. Determine, with proof, the smallest positive multiple of 99 all of whose digits are either 1 or 2.
2017 Irish Math Olympiad
Problem 2

2. Solve the equations : 

$$a + b + c = 0 , a^2 + b^2 + c^2 = 1 , a^3 + b^3 +c^3 = 4abc $$for $ a,b,$ and $c. $
2017 Irish Math Olympiad
Problem 3

3. Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.
2017 Irish Math Olympiad
Problem 4

4. An equilateral triangle of integer side length $n \geq 1$ is subdivided into small triangles of unit side length, as illustrated in the figure below for $n = 5$. (figure to be added) In this diagram. A subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines.



It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color.

Let $f(n)$ denote the number of distinct colorings satisfying this condition.



Determine, with proof, $f(n)$ for every $n \geq 1$
2017 Irish Math Olympiad
Problem 5

5. The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and

$$a_{n+2} = 2a_{n+1} + 41a_n$$Prove that $a_{2016}$ is divisible by $2017.$
2017 Irish Math Olympiad
Problem 6 (Start of Paper 2, time for paper : 3 hours)

6. Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?
2017 Irish Math Olympiad
Problem 7

7. 5 teams play in a soccer competition where each team plays one match against each of the other four teams. A winning team gains $5$ points and a losing team $0$ points. For a $0-0$ draw both teams gain $1$ point, and for other draws ($1-1,2-2,3-3,$etc.) both teams gain 2 points. At the end of the competition, we write down the total points for each team, and we find that they form 5 consecutive integers. What is the minimum number of goals scored?
2017 Irish Math Olympiad
Problem 8

8. A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that :

$$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$
2017 Irish Math Olympiad
Problem 9

9. Show that for all non-negative numbers $a,b$,

$$ 1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10} $$When is equality attained?
2017 Irish Math Olympiad
Problem 10 (The finale)

Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies

$$(\sum_{k=1}^{n} a_k )^{m}  = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$.

(a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero.

(b) Find a sequence none of whose terms are zero but which is $2017$-powerful
2018 Irish Math Olympiad
Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game.

Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.
2018 Irish Math Olympiad
The triangle $ABC$ is right-angled at $A$. Its incentre is $I$, and $H$ is the foot of the perpendicular from $I$ on $AB$. The perpendicular from $H$ on $BC$ meets $BC$ at $E$, and it meets the bisector of $\angle ABC$ at $D$. The perpendicular from $A$ on $BC$ meets $BC$ at $F$. Prove that $\angle EFD = 45^o$
2018 Irish Math Olympiad
Find all functions $f(x) = ax^2 + bx + c$, with $a \ne 0$, such that $f(f(1)) = f(f(0)) = f(f(-1))$ .
2018 Irish Math Olympiad
We say that a rectangle with side lengths $a$ and $b$ fits inside a rectangle with side lengths $c$ and $d$ if either ($a  \le c$ and $b  \le  d$) or ($a  \le  d$ and $b \le c$). For instance, a rectangle with side lengths $1$ and $5$ fits inside another rectangle with side lengths $1$ and $5$, and also fits inside a rectangle with side lengths $6$ and $2$.

Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$, $B$, and $C$ in $S$ such that $A$ fits inside $B$, and $B$ fits inside $C$.
2018 Irish Math Olympiad
Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$. A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$. The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$.
2018 Irish Math Olympiad
Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.
2018 Irish Math Olympiad
Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$
2018 Irish Math Olympiad
Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$. The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$. The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$, and $|MD| = |ME|$. The point $F$ is the intersection of $MD$ and $AB$. Prove that $\angle BFM = \angle BME$.
2018 Irish Math Olympiad
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$  for $n \ge 1$.

Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
2018 Irish Math Olympiad
The game of Greed starts with an initial configuration of one or more piles of stones.

Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$. At each turn, a player has two choices:

•  take one stone from any one of the piles (a simple move);

•  take one stone from each of the remaining piles (a greedy move).

The player who takes the last stone wins.

Consider the following two initial configurations:

(a) There are $2018$ piles, with either $20$ or $18$ stones in each pile.

(b) There are four piles, with $17, 18, 19$, and $20$ stones, respectively.

In each case, find an appropriate strategy that guarantees victory to one of the players.
2021 Irish Math Olympiad
Let $N = 15! = 15\cdot 14\cdot 13  ... 3\cdot 2\cdot 1$. Prove that $N$ can be written as a product of nine different integers all between $16$ and $30$ inclusive.
2021 Irish Math Olympiad
An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot  |CG|$.
2021 Irish Math Olympiad
For each integer $n \ge 100$ we define $T(n)$ to be the number obtained from $n$ by moving the two leading digits to the end. For example, $T(12345) = 34512$ and $T(100) = 10$. Find all integers $n \ge  100$ for which $n + T(n) = 10n$.
2021 Irish Math Olympiad
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.

What is the remainder when $m$ is divided by $19$?
2021 Irish Math Olympiad
The function $g : [0, \infty) \to  [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge  0$.

You are also told that for $2 \le  x \le 3$: $g(x) = \frac{x + 1}{2}$.

(a) Find $g(2021)$.

(b) Find $g(1/2021)$.
2021 Irish Math Olympiad
A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.
2021 Irish Math Olympiad
Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$. He then performs the same calculation on the red cells to compute $S_R$.

If S_B-  S_R = 50, determine (with proof) all possible values of $k$.