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[Solutions] India Regional Mathematical Olympiad 2012

Region 1

  1. Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $ \Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $ \Gamma$.
  2. Let $a,b,c$ be positive integers such that $a|b^5, b|c^5$ and $c|a^5$. Prove that $abc|(a+b+c)^{31}$.
  3. Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.
  4. Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.
  5. Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.
  6. Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

Region 2

  1. Let $ABCD$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B$; $D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A$; $C$ as end points of the arc. Inscribe a circle touching the arc $AC$ internally, the arc $BD$ internally and also touching the side $AB$. Find the radius of the circle?. 
  2. Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $$abc|(a+b+c)^{21}$$
  3. Same as R1 P3
  4. Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$. 
  5. Let $ABC$ be a triangle. Let $D$, $E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$. 
  6. Show that for all real numbers $x,y,z$ such that $x + y + z = 0$ and $xy + yz + zx = -3$, the expression $x^3y + y^3z + z^3x$ is a constant.

Region 3

  1. Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B$, $D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A$, $C$ as end points of the arc. Inscribe a circle $\Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $\Gamma$.
  2. Let $a,b,c$ be positive integers such that $a|b^5, b|c^5$ and $c|a^5$. Prove that $abc|(a+b+c)^{31}$.
  3. Same as R1 P3
  4. Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.
  5. Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.
  6. Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

Region 4

  1. Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ externally, the arc $BD$ externally and also touching the side $AD$. Find the radius of $\Gamma$.
  2. Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$
  3. Same as R1 P3
  4. Same as R2 P4
  5. Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$.
  6. Solve the system of equations for positive real numbers $$\frac{1}{xy}=\frac{x}{z}+ 1,\quad \frac{1}{yz} = \frac{y}{x} + 1,\quad \frac{1}{zx} =\frac{z}{y}+ 1$$

Region 5

  1. Find with proof all non–zero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.
  2. Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.
  3. Solve for real $x$ $$2^{2x} \cdot  2^{3\{x\}} = 11 \cdot  2^{5\{x\}} + 5  \cdot 2^{2[x]}$$ (For a real number $x, [x]$ denotes the greatest integer less than or equal to $x$. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)
  4. $H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.
  5. Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\dfrac{1}{a}+ \dfrac{2}{b} +\dfrac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.
  6. Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$. We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$. Define WSUM for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if $A = \{2, 5, 7, 8\}$, WSUM is $3(2 + 7) + 2(5 + 8)$.) Find the sum of WSUMs over all the subsets of $S$. (Assume that WSUM for the null set is $0$.)
  7. On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.
  8. Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$. Prove that $$\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}.$$

Region 6

  1. Let $ABCD$ be a convex quadrilateral such that $\angle ADC=\angle BCD>90^{\circ}$. Let $E$ be the point of intersection of $AC$ and the line through $B$ parallel to $AD;$ let $F$ be the point of intersection of $BD$ and the line through $A$ parallel to $BC.$ Prove that $EF\parallel CD.$
  2. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$
  3. Find all natural numbers $x,y,z$ such that \[(2^x-1)(2^y-1)=2^{2^z}+1.\]
  4. Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.
  5. Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$
  6. A computer program generated $175$ positive integers at random, none of which had a prime divisor grater than $10.$ Prove that there are three numbers among them whose product is the cube of an integer.

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MOlympiad: [Solutions] India Regional Mathematical Olympiad 2012
[Solutions] India Regional Mathematical Olympiad 2012
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