Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 Irish Math Olympiad
$P$, $Q$, $R$ and $S$ are (distinct) points on a circle. $PS$ is a diameter and $QR$ is parallel to the diameter $PS$. $PR$ and $QS$ meet at $A$. Let $O$ be the centre of the circle and let $B$ be chosen so that the quadrilateral $POAB$ is a parallelogram. Prove that $BQ$ = $BP$ .
2003 Irish Math Olympiad
For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate $$\sum_{k=1}^{2003} (a_k-b_k).$$
2003 Irish Math Olympiad
Eight players, Ann, Bob, Con, Dot, Eve, Fay, Guy and Hal compete in a chess tournament. No pair plays together more than once and there is no group of five people in which each one plays against all of the other four.



(a) Write down an arrangement for a tournament of $24$ games satisfying these conditions.



(b) Show that it is impossible to have a tournament of more than $24$ games satisfying these conditions.
2003 Irish Math Olympiad
show that thee is no   function f definedonthe positive real numbes such that :

$f(y) > (y-x)f(x)^2$
2003 Irish Math Olympiad
If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that:



a) $abc+\frac{28}{27}\geq ab+bc+ac$;



b) $abc+1<ab+bc+ac$





See also   (problem 1)



:)
2003 Irish Math Olympiad
$\ ABCD$ is a quadrilateral. the feet of the perpendicular from $\ D$ to $\ AB, BC$ are $\ P,Q$ respectively, and the feet of the perpendicular from $\ B$ to $\ AD,CD$ are $\ R,S$ respectively. Show that if $\angle PSR= \angle SPQ$, then $\ PR=QS$.
2003 Irish Math Olympiad
Find all the (x,y) integer ,if

$y^2+2y=x^4+20x^3+104x^2+40x+2003$
2003 Irish Math Olympiad
Given  real positive  a,b , find  the larget  real  c  such  that  $c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})$  for all positive ral  x.



There is a  solution  here,,,,



but im  wondering if there is a  better one .



Thank  you.
2003 Irish Math Olympiad
(a) In how many ways can $1003$ distinct integers be chosen from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers differ by $10?$

(b) Show that there are $(3(5151) + 7(1700)) 101^7$ ways to choose $1002$ distinct integers from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers differ by $10.$
2004 Irish Math Olympiad
1. (a) For which positive integers n, does 2n divide the sum of the first n positive

integers?

(b) Determine, with proof, those positive integers n (if any) which have the

property that 2n + 1 divides the sum of the first n positive integers.
2004 Irish Math Olympiad
Each of the players in a tennis tournament played one match against each of

the others. If every player won at least one match, show that there is a group

A; B; C of three players for which A beat B, B beat C and C beat A.
2004 Irish Math Olympiad
$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.
2004 Irish Math Olympiad
Prove that there are only two real numbers $x$ such that $$(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) = 720$$
2004 Irish Math Olympiad
Let $a,b\ge 0$. Prove that $$\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right)\le 3(a^2+b^2)$$with equality if and only if $a=b$.
2004 Irish Math Olympiad
Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.
2004 Irish Math Olympiad
$A$ and $B$ are distinct points on a circle $T$. $C$ is a point distinct from $B$ such that $|AB|=|AC|$, and such that $BC$ is tangent to $T$ at $B$. Suppose that the bisector of $\angle ABC$ meets $AC$ at a point $D$ inside $T$. Show that $\angle ABC>72^\circ$.
2004 Irish Math Olympiad
Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number $$\sqrt[3]{n^{3}+2n^{2}+n}$$
2004 Irish Math Olympiad
Define the function $m$ of the three real variables $x$, $y$, $z$ by $m$($x$,$y$,$z$) = max($x^2$,$y^2$,$z^2$), $x$, $y$, $z$ ∈ $R$.



Determine, with proof, the minimum value of $m$ if $x$,$y$,$z$ vary in $R$ subject to the following restrictions:



$x$ + $y$ + $z$ = 0,

$x^2$ + $y^2$ + $z^2$ = 1.
2004 Irish Math Olympiad
Suppose $p,q$ are distinct primes and $S$ is a subset of $\{1,2,\dots ,p-1\}$. Let $N(S)$ denote the number of solutions to the equation $$\sum_{i=1}^{q}x_i\equiv 0\mod p$$where $x_i\in S$, $i=1,2,\dots ,q$. Prove that $N(S)$ is a multiple of $q$.
2005 Irish Math Olympiad
Show that $ 2005^{2005}$ is a sum of two perfect squares, but not a sum of two perfect cubes.
2005 Irish Math Olympiad
Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.
2005 Irish Math Olympiad
Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.
2005 Irish Math Olympiad
Determine the number of arrangements $ a_1,a_2,...,a_{10}$ of the numbers $ 1,2,...,10$ such that $ a_i>a_{2i}$ for $ 1 \le i \le 5$ and $ a_i>a_{2i+1}$ for $ 1 \le i \le 4$.
2005 Irish Math Olympiad
Let $ a,b,c$ be nonnegative real numbers. Prove that:



$ \frac{1}{3}((a-b)^2+(b-c)^2+(c-a)^2) \le a^2+b^2+c^2-3 \sqrt[3]{a^2 b^2 c^2 } \le (a-b)^2+(b-c)^2+(c-a)^2.$
2005 Irish Math Olympiad
Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.
2005 Irish Math Olympiad
Using the digits: $ 1,2,3,4,5,$ players $ A$ and $ B$ compose a $ 2005$-digit number $ N$ by selecting one digit at a time: $ A$ selects the first digit, $ B$ the second, $ A$ the third and so on. Player $ A$ wins if and only if $ N$ is divisible by $ 9$. Who will win if both players play as well as possible?
2005 Irish Math Olympiad
Let $ x$ be an integer and $ y,z,w$ be odd positive integers. Prove that $ 17$ divides $ x^{y^{z^w}}-x^{y^z}$.
2005 Irish Math Olympiad
Find the first digit to the left and the first digit to the right of the decimal point in the expansion of $ (\sqrt{2}+\sqrt{5})^{2000}.$
2005 Irish Math Olympiad
Suppose that $ m$ and $ n$ are odd integers such that $ m^2-n^2+1$ divides $ n^2-1$. Prove that $ m^2-n^2+1$ is a perfect square.
2006 Irish Math Olympiad
Are there integers $x,y$ and $z$ which satisfy the equation $$z^2=(x+1)(y^2-1)+n$$when (a) $n=2006$ (b) $n=2007$?
2006 Irish Math Olympiad
$P$ and $Q$ are points on the equal sides $AB$ and $AC$ respectively of an isosceles triangle $ABC$ such that $AP=CQ$. Moreover, neither $P$ nor $Q$ is a vertex of $ABC$. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of the triangle $ABC$.
2006 Irish Math Olympiad
Prove that a square of side 2.1 units can be completely covered by seven squares of side 1 unit.



Extra: Try to prove that 7 is the minimal amount.
2006 Irish Math Olympiad
Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$, $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$
2006 Irish Math Olympiad
Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.
2006 Irish Math Olympiad
The rooms of a building are arranged in a $m\times n$ rectangular grid (as shown below for the $5\times 6$ case). Every room is connected by an open door to each adjacent room, but the only access to or from the building is by a door in the top right room. This door is locked with an elaborate system of $mn$ keys, one of which is located in every room of the building. A person is in the bottom left room and can move from there to any adjacent room. However, as soon as the person leaves a room, all the doors of that room are instantly and automatically locked. Find, with proof, all $m$ and $n$ for which it is possible for the person to collect all the keys and escape the building.





[asy]

unitsize(5mm);

defaultpen(linewidth(.8pt));

fontsize(25pt);

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

{

for(int j=0; j<= 6; ++j)

{

draw((0,i)--(9,i));

draw((1.5*j,0)--(1.5*j,5));

}}

dot((.75, .5));

label("$\ast$",(8.25,4.5));



dot((11, 3));

label("$\ast$",(11,1.75));

label("room with locked external door",(18,1.9));

label("starting position",(15.3,3));



[/asy]
2006 Irish Math Olympiad
$ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$.
2006 Irish Math Olympiad
let x,y are positive and $ \in R$ that : $ x+2y=1$.prove that :

$$ \frac{1}{x}+\frac{2}{y} \geq \frac{25}{1+48xy^2}$$
2006 Irish Math Olympiad
Let $n$ be a positive integer.

Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.
2006 Irish Math Olympiad
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.



Proposed by Horst Sewerin, Germany
2007 Irish Math Olympiad
Find all prime numbers $ p$ and $ q$ such that $ p$ divides $ q+6$ and $ q$ divides $ p+6$.
2007 Irish Math Olympiad
Prove that the triangle ABC is right-angled if it holds: $$ \sin^2 A+\sin^2 B+\sin^2 C = 2 $$
2007 Irish Math Olympiad
The point $ P$ is a fixed point on a circle and $ Q$ is a fixed point on a line. The point $ R$ is a variable point on the circle such that $ P,Q,$ and $ R$ are not collinear. The circle through $ P,Q,$ and $ R$ meets the line again at $ V$. Show that the line $ VR$ passes through a fixed point.
2007 Irish Math Olympiad
Air Michael and Air Patrick operate direct flights connecting Belfast, Cork, Dublin, Galway, Limerick, and Waterord. For each pair of cities exactly one of the airlines operates the route (in both directions) connecting the cities. Prove that there are four cities for which one of the airlines operates a round trip. (Note that a round trip of four cities $ P,Q,R,$ and $ S$, is a journey that follows the path $ P \rightarrow Q \rightarrow R \rightarrow S \rightarrow P$.)
2007 Irish Math Olympiad
Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$.



$ (a)$ Prove that: $ \frac{n+1-2r}{n+1-r} \binom{n}{r}$ is an integer.



$ (b)$ Prove that: $ \displaystyle\sum_{r=0}^{[n/2]}\frac{n+1-2r}{n+1-r} \binom{n}{r}<2^{n-2}$ for all $ n \ge 9$.
2007 Irish Math Olympiad
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)=x^3-2007x+2002.$



Determine the value of: $ \frac{r-1}{r+1}+\frac{s-1}{s+1}+\frac{t-1}{t+1}$.
2007 Irish Math Olympiad
Suppose that $ a,b,$ and $ c$ are positive real numbers. Prove that:



$ \frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \le \frac {\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}}{3}$.



For each of the inequalities, find the conditions on $ a,b,$ and $ c$ such that equality holds.
2007 Irish Math Olympiad
Let $ ABC$ be a triangle the lengths of whose sides $ BC,CA,AB,$ respectively, are denoted by $ a,b,$ and $ c$. Let the internal bisectors of the angles $ \angle BAC, \angle ABC, \angle BCA,$ respectively, meet the sides $ BC,CA,$ and $ AB$ at $ D,E,$ and $ F$. Denote the lengths of the line segments $ AD,BE,CF$ by $ d,e,$ and $ f$, respectively. Prove that:



$ def=\frac{4abc(a+b+c) \Delta}{(a+b)(b+c)(c+a)}$ where $ \Delta$ stands for the area of the triangle $ ABC$.
2007 Irish Math Olympiad
Find the number of zeros in which the decimal expansion of $ 2007!$ ends. Also find its last non-zero digit.
2007 Irish Math Olympiad
Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)=x^2+ax+b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)=\frac{g(x)}{h(x)}$ for all real numbers $ x$.
2008 Irish Math Olympiad
Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations



$ 2p_1 + 3p_2 + 5p_3 + 7p_4 = 162$

$ 11p_1 + 7p_2 + 5p_3 + 4p_4 = 162$



Find all possible values of the product $ p_1p_2p_3p_4$
2008 Irish Math Olympiad
For positive real numbers $ a$, $ b$, $ c$ and $ d$ such that $ a^2 + b^2 + c^2 + d^2 = 1$  prove that



$ a^2b^2cd + +ab^2c^2d + abc^2d^2 + a^2bcd^2 + a^2bc^2d + ab^2cd^2 \le 3/32,$



and determine the cases of equality.
2008 Irish Math Olympiad
Determine, with proof, all integers $ x$ for which $ x(x+1)(x+7)(x+8)$ is a perfect square.
2008 Irish Math Olympiad
How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$?
2008 Irish Math Olympiad
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| = |AB|$.

Prove that $ |AD|^2 = |AB|.|BC|$ if and only if $ \angle CBD = 30^\circ$.
2008 Irish Math Olympiad
Find, with proof, all triples of integers $ (a,b,c)$ such that $ a, b$ and $ c$ are the lengths of the sides of a right angled triangle whose area is $ a + b + c$
2008 Irish Math Olympiad
Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| = |PB|$.
2008 Irish Math Olympiad
Find $ a_3,a_4,...,a{}_2{}_0{}_0{}_8$, such that $ a_i =\pm1$ for $ i=3,...,2008$ and



$ \sum\limits_{i=3}^{2008} a_i2^i = 2008$



and show that the numbers $ a_3,a_4,...,a_{2008}$ are uniquely determined by these conditions.
2008 Irish Math Olympiad
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [-1,0,1]$ for $ i=1,...,n,$ and



$ x_1+...+x_n \equiv k$ mod 4



a) Prove that $ f_1(n) = f_3(n)$ for all positive integers $ n$.



(b) Prove that



$ f_0(n) = [{3^n + 2 + [-1]^n}] / 4$



for all positive integers $ n$.
2008 Irish Math Olympiad
Suppose that $ x, y$ and $ z$ are positive real numbers such that $ xyz \ge 1$.



(a) Prove that

$ 27 \le (1 + x + y)^2 + (1+ x + z)^2 + (1 + y + z)^2$,

with equality if and only if $ x = y = z = 1$.



(b) Prove that

$ (1 + x + y)^2 + (1+ x + z)^2 + (1 + y + z)^2$ $ \le 3(x + y + z)^2$,

with equality if and only if $ x = y = z = 1$.
2009 Irish Math Olympiad
Hamilton Avenue has eight houses. On one side of the street are the houses

numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An

eccentric postman starts deliveries at house 1 and delivers letters to each of

the houses, finally returning to house 1 for a cup of tea. Throughout the

entire journey he must observe the following rules. The numbers of the houses

delivered to must follow an odd-even-odd-even pattern throughout, each house

except house 1 is visited exactly once (house 1 is visited twice) and the postman

at no time is allowed to cross the road to the house directly opposite. How

many different delivery sequences are possible?
2009 Irish Math Olympiad
Let $ABCD$ be a square. The line segment $AB$ is divided internally at $H$ so that $|AB|\cdot |BH|=|AH|^2$. Let $E$ be the midpoints of $AD$ and $X$ be the midpoint of $AH$. Let $Y$ be a point on $EB$ such that $XY$ is perpendicular to $BE$. Prove that $|XY|=|XH|$.
2009 Irish Math Olympiad
Find all positive integers $n$ for which $n^8+n+1$ is a prime number.
2009 Irish Math Olympiad
Given an $n$-tuple of numbers $(x_1,x_2,\dots ,x_n)$ where each $x_i=+1$ or $-1$, form a new $n$-tuple $$(x_1x_2,x_2x_3,x_3x_4,\dots ,x_nx_1),$$and continue to repeat this operation. Show that if $n=2^k$ for some integer $k\ge 1$, then after a certain number of repetitions of the operation, we obtain the $n$-tuple $$(1,1,1,\dots ,1).$$
2009 Irish Math Olympiad
Hello.



Suppose $a$, $b$, $c$ are real numbers such that $a+b+c = 0$ and $a^{2}+b^{2}+c^{2} = 1$.

Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$ and determine the cases of equality.
2009 Irish Math Olympiad
Let $P(x)$ be a polynomial with rational coefficients. Prove that there exists a positive integer $n$ such that the polynomial $Q(x)$ defined by

$$Q(x)= P(x+n)-P(x)$$

has integer coefficients.
2009 Irish Math Olympiad
For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$
2009 Irish Math Olympiad
Find all pairs $(a,b)$ of positive integers such that $(ab)^2 - 4(a+b)$ is the square of an integer.
2009 Irish Math Olympiad
At a strange party, each person knew exactly $22$ others.

For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew.

For any pair of people $X$ and $Y$ who did not know one another, there were exactly $6$ other people that they both knew.

How many people were at the party?
2009 Irish Math Olympiad
In the triangle $ABC$ we have $|AB|<|AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$.



Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$
2010 Irish Math Olympiad
Find the least $k$ for which the number $2010$ can be expressed as the sum of the squares of $k$ integers.
2010 Irish Math Olympiad
Let $ABC$ be a triangle and let $P$ denote the midpoint of the side $BC$. Suppose that there exist two points $M$ and $N$ interior to the side $AB$ and $AC$ respectively, such that $$|AD|=|DM|=2|DN|,$$where $D$ is the intersection point of the lines $MN$ and $AP$. Show that $|AC|=|BC|$.
2010 Irish Math Olympiad
Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that

(a) $xy+yz+xz\ge 9xyz$;

(b) $xy+yz+xz<\frac{1}{4}+3xyz$;
2010 Irish Math Olympiad
The country of Harpland has three types of coins: green, white and orange. The unit of currency in Harpland is the shilling. Any coin is worth a positive integer number of shillings, but coins of the same color may be worth different amounts. A set of coins is stacked in the form of an equilateral triangle of side $n$ coins, as shown below for the case of $n=6$.



[asy]

size(100);

for (int j=0; j<6; ++j)

{

for (int i=0; i<6-j; ++i)

{

draw(Circle((i+j/2,0.866j),0.5));

}

}

[/asy]



The stacking has the following properties:



(a) no coin touches another coin of the same color;



(b) the total worth, in shillings, of the coins lying on any line parallel to one of the sides of the triangle is divisible by by three.



Prove that the total worth in shillings of the green coins in the triangle is divisible by three.
2010 Irish Math Olympiad
Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.
2010 Irish Math Olympiad
There are $14$ boys in a class. Each boy is asked how many other boys in the class have his first name, and how many have his last name. It turns out that each number from $0$ to $6$ occurs among the answers.



Prove that there are two boys in the class with the same first name and the same last name.
2010 Irish Math Olympiad
For each odd integer $p\ge 3$ find the number of real roots of the polynomial $$f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.$$
2010 Irish Math Olympiad
In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.
2010 Irish Math Olympiad
Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$Prove that $a_n$ is not divisible by $(n-1)^2$.
2010 Irish Math Olympiad
Suppose $a,b,c$ are the side lengths of a triangle $ABC$. Show that $$x=\sqrt{a(b+c-a)}, y=\sqrt{b(c+a-b)}, z=\sqrt{c(a+b-c)}$$are the side lengths of an acute-angled triangle $XYZ$, with the same area as $ABC$, but with a smaller perimeter, unless $ABC$ is equilateral.
2011 Irish Math Olympiad
Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$
2011 Irish Math Olympiad
Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$, respectively, the lengths of the medians which connect $A,B,C$, respectively, with the centers of the corresponding opposite sides.



(a) Prove that $2m_a<b+c$. Deduce that $m_a+m_b+m_c<a+b+c$.

(b) Give an example of

(i) a triangle in which $m_a>\sqrt{bc}$;

(ii) a triangle in which $m_a\le \sqrt{bc}$.
2011 Irish Math Olympiad
The integers $a_0, a_1, a_2, a_3,\ldots$ are defined as follows:

$a_0 = 1$, $a_1 = 3$, and $a_{n+1} = a_n + a_{n-1}$ for all $n \ge 1$.

Find all integers $n \ge 1$ for which $na_{n+1} + a_n$ and $na_n + a_{n-1}$ share a common factor greater than $1$.
2011 Irish Math Olympiad
The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$. Prove that



(a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$.



(b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$.
2011 Irish Math Olympiad
In the mathematical talent show called “The $X^2$-factor” contestants are scored by a a panel of $8$ judges. Each judge awards a score of $0$ (‘fail’), $X$ (‘pass’), or $X^2$ (‘pass with distinction’). Three of the contestants were Ann, Barbara and David. Ann was awarded the same score as Barbara by exactly $4$ of the judges. David declares that he obtained different scores to Ann from at least $4$ of the judges, and also that he obtained different scores to Barbara from at least $4$ judges.



In how many ways could scores have been allocated to David, assuming he is telling the truth?
2011 Irish Math Olympiad
Prove that $$\frac{2}{3}+\frac{4}{5}+\dots +\frac{2010}{2011}$$is not an integer.
2011 Irish Math Olympiad
In a tournament with $n$ players, $n$ < 10, each player plays once against each other player scoring 1 point for a win and 0 points for a loss. Draws do not occur. In a particular tournament only one player ended with an odd number of points and was ranked fourth. Determine whether or not this is possible. If so, how many wins did the player have?
2011 Irish Math Olympiad
$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$.
2011 Irish Math Olympiad
Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$Prove that

(i)

$$\frac{3}{4}\le xy+yz+zx<1$$(ii)

$$xyz\le \frac{1}{8}$$Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$and derive the case of equality.
2011 Irish Math Olympiad
Find with proof all solutions in nonnegative integers $a,b,c,d$ of the equation $$11^a5^b-3^c2^d=1$$