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[Solutions] India Regional Mathematical Olympiad 2014

Region 1

  1. In acute $\triangle ABC,$ let $D$ be the foot of perpendicular from $A$ on $BC$. Consider points $K, L, M$ on segment $AD$ such that $$AK= KL= LM= MD.$$ Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that $BD= DC$.
  2. Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let $$\sum_{i=1}^{n}a_{2i-1}^2 =x,\,\sum _{i=1}^{n}a_{2i}^2=y,\,a_n+a_{n+1}=z.$$ Express $d$ in terms of $x,y,z,n$.
  3. Suppose for some positive integers $r$ and $s$, $2^r$ is obtained by permuting the digits of $2^s$ in decimal expansion. Prove that $r=s$.
  4. Is it possible to write the numbers $17$,$18$,$19$,...$32$ in a $4*4$ grid of unit squares with one number in each square such that if the grid is divided into four $2*2$ subgrids of unit squares ,then the product of numbers in each of the subgrids divisible by $16$?
  5. Let $ABC$ be an acute angled triangle with $H$ as its orthocentre. For any point $P$ on the circumcircle of triangle $ABC$, let $Q$ be the point of intersection of the line $BH$ with line $AP$. Show that there is a unique point $X$ on the circumcircle of triangle $ABC$ such that for every $P$ other than $B,C$, the circumcircle of $HPQ$ passes through $X$.
  6. Let $x_1,x_2,x_3 \ldots x_{2014}$ be positive real numbers such that $\sum_{j=1}^{2014} x_j=1$. Determine with proof the smallest constant $K$ such that \[K\sum_{j=1}^{2014}\frac{x_j^2}{1-x_j} \ge 1\]

Region 2

  1. In an acute-angled triangle $ABC$, $\angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$.
  2. Let $x, y, z$ be positive real numbers. Prove that $$\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z).$$
  3. Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.
  4. For any positive integer $ n > 1$, let $P(n)$ denote the largest prime not exceeding $n$. Let $N(n)$ denote the next prime larger than $P(n)$. (For example $P(10) = 7$ and $N(10) = 11$, while $P(11) = 11$ and $N(11) = 13$.) If $n + 1$ is a prime number, prove that the value of the sum $$\frac{1}{P(2)N(2)} + \frac{1}{P(3)N(3)} + \cdot\cdot\cdot + \frac{1}{P(n)N(n)} = \frac{n-1}{2n+2}.$$
  5. Let $ABC$ be a triangle with $AB > AC$. Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$. Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$. Prove that $BQ = 2AP.$
  6. Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$. Let $r_j$ and $c_k$ denote the product of all numbers in $j$-th row and $k$-th column respectively, $1 \le j, k \le n$. Prove that $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

Region 3

  1. Let $ABC$ be a triangle with $\angle ABC $ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$. Prove that $RX$ is perpendicular to $BC$.
  2. Find all real $x,y$ such that \[x^2 + 2y^2 + \frac{1}{2} \le x(2y+1) \]
  3. Prove that for any natural number $n < 2310 $, $n(2310-n)$ is not divisible by $2310$.
  4. Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014}\\ 2y-2z +\dfrac1x = \dfrac1{2014} \\ 2z-2x+ \dfrac1y = \dfrac1{2014}.\]
  5. Let $ABC$ be a triangle and let $X$ be on $BC$ such that $AX=AB$. let $AX$ meet circumcircle $\omega$ of triangle $ABC$ again at $D$. prove that circumcentre of triangle $BDX$ lies on $\omega$. 
  6. For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$.

Region 4

  1. Let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$. Let $CE$ be the perpendicular from $C$ on $AB$. Prove that $ CE^2 = AB\cdot CD $.
  2. Let $x,y$ be positive real numbers. Prove that $$4x^4+4y^3+5x^2+y+1\geq 12xy.$$
  3. Let $m,n$ be natural number with $m>n$. Find all such pairs of $(m,n)$ such that $$\gcd(n+1,m+1) = \gcd(n+2,m+2) =\ldots = \gcd(m, 2m-n) = 1.$$
  4. Let $ABC$ be a right angled triangle with inradius $1$. Find the minimum area of triangle $ABC$ 
  5. Let $ABC$ be a triangle and $I$ be its incentre. let the incircle of $ABC$ touch $BC$ at $D$. Let incircle of triangle $ABD$ touch $AB$ at $E$ and incircle of triangle $ACD$ touch $BC$ at $F$. Prove that $B,E,I,F$ are concyclic.
  6. In the adjacent figure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?.

Mumbai Region

  1. Three positive real numbers $a,b,c$ are such that $$a^2+5b^2+4c^2-4ab-4bc=0.$$ Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.
  2. The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.
  3. Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.
  4. A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules
    a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$;
    b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$.
    How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?
  5. Let $a,b,c$ be positive real numbers such that \[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \] Prove that $$(1+a^2)(1+b^2)(1+c^2)\ge 125.$$ When does equality hold?
  6. Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE$, $BDF$, $CED$, respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.

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MOlympiad: [Solutions] India Regional Mathematical Olympiad 2014
[Solutions] India Regional Mathematical Olympiad 2014
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