RMO 2013

Let $ABC$ be an acute-angled triangle. The circle $\Gamma$ with $BC$ as diameter intersects $AB$ and $AC$ again at $P$ and $Q$ , respectively. Determine $\angle BAC$ given that the orthocenter of triangle $APQ$ lies on $\Gamma$ .

Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$ , where $a,b,c$ are integers with $c\not=0$ . Suppose that the following conditions hold:

[list=a][] $f(1)=0$ , []the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$ .[/list]

Find the value of $a^{2013}+b^{2013}+c^{2013}$ .

Find all primes $p$ and $q$ such that $p$ divides $q^2-4$ and $q$ divides $p^2-1$ .

Find the number of $10$ -tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and $$a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.$$

Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$ .

Prove that there do not exist natural numbers $x$ and $y$ with $x>1$ such that , $$\frac{x^7-1}{x-1}=y^5+1$$

In a triangle $ABC$ , $AD$ is the altitude from $A$ , and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $BH$ at $H$ . Prove that the line $DK$ bisects $AC$ .

Consider the expression $$2013^2+2014^2+2015^2+ \cdots+n^2$$ Prove that there exists a natural number $n > 2013$ for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal $9999$

Let $n \ge 3$ be a natural number and let $P$ be a polygon with $n$ sides. Let $a_1,a_2,\cdots, a_n$ be the lengths of sides of $P$ and let $p$ be its perimeter. Prove that $$\frac{a_1}{p-a_1}+\frac{a_2}{p-a_2}+\cdots + \frac{a_n}{p-a_n} < 2$$

For a natural number $n$ , let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$ .

Find the number of eight-digit numbers the sum of whose digits is $4$

Find all $4$ -tuples $(a,b,c,d)$ of natural numbers with $a \le b \le c$ and $a!+b!+c!=3^d$

In an acute-angled triangle $ABC$ with $AB < AC$ , the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$ . Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$ .

A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$ . Prove that $f(x)+2x$ is not a fermat polynomial

Let $ABC$ be a triangle which it not right-angled. Define a sequence of triangles $A_iB_iC_i$ , with $i \ge 0$ , as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$ , $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$ , $C_iA_i$ , $A_iB_i$ , respectively. Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$ . Prove that $\angle A = 60^{\circ}$ .

Let $\omega$ be a circle with centre $O$ . Let $\gamma$ be another circle passing through $O$ and intersecting $\omega$ at points $A$ and $B$ . $A$ diameter $CD$ of $\omega$ intersects $\gamma$ at a point $P$ different from $O$ . Prove that $\angle APC= \angle BPD$

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

Given real numbers $a,b,c,d,e>1$ . Prove that $$\frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge 20$$

Let $x$ be a non-zero real numbers such that $x^4+\frac{1}{x^4}$ and $x^5+\frac{1}{x^5}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is a rational number.

In a triangle $ABC$ , let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$ . The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$ , and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$ . Prove that $H$ is the incentre of triangle $PQR$ .

Suppose that the vertices of a regular polygon of $20$ sides are coloured with three colours - red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices $A,B,C$ of the polygon having the same colour such that triangle $ABC$ is isosceles.

Let $ABC$ be an isosceles triangle with $AB=AC$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $AB$ of $\Gamma$ not containing $C$ . A point $E$ is on arc $AC$ of $\Gamma$ not containing $B$ . If $AD=CE$ prove that $BE$ is parallel to $AD$ .

Find all triples $(p,q,r)$ of primes such that $pq=r+1$ and $2(p^2+q^2)=r^2+1$ .

A finite non-empty set of integers is called $3$ -good if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\{0,1,2,\ldots,9\}$ .

In a triangle $ABC$ , points $D$ and $E$ are on segments $BC$ and $AC$ such that $BD=3DC$ and $AE=4EC$ . Point $P$ is on line $ED$ such that $D$ is the midpoint of segment $EP$ . Lines $AP$ and $BC$ intersect at point $S$ . Find the ratio $BS/SD$ .

Let $a_1,b_1,c_1$ be natural numbers. We define $$a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\, b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\, c_2=\gcd(a_1,b_1),$$ and $$a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\, b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\, c_3=\operatorname{lcm}(a_2,b_2).$$ Show that $\gcd(b_3,c_3)=a_2$ .

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$ , where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$ . Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$ .