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[Answer Keys] India Pre-Regional Mathematical Olympiad 2014

  1. A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$? 
  2. The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence? 
  3. Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$? 
  4. In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle? 
  5. If real numbers $a, b, c, d, e$ satisfy $$a + 1 = b + 2 = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3,$$ what is the value of $a^2 + b^2 + c^2 + d^2 + e^2$ ? 
  6. What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots? 
  7. If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ? 
  8. Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have? 
  9. Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\dfrac1b$ and $b + \dfrac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$? 
  10. In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$? 
  11. For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$? 
  12. Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ? 
  13. For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\dfrac{8n}{9999-n}$ an integer? 
  14. One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1/7$-th of the total amount of milk and $2/17$-th of the total amount of coffee. How many people are there in Manjul’s family? 
  15. Let $XOY$ be a triangle with $\angle XOY = 90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Suppose that $XN = 19$ and $YM =22$. What is $XY$? 
  16. In a triangle $ABC$, let $I$ denote the incenter. Let the lines $AI$, $BI$ and $CI$ intersect the incircle at $P$, $Q$ and $R$, respectively. If $\angle BAC = 40^\circ$, what is the value of $\angle QPR$ in degrees ? 
  17. For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$? 
  18. Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f (999)$ ? 
  19. Let $x_1,x_2,... ,x_{2014}$ be real numbers different from $1$, such that $$x_1 + x_2 +...+x_{2014} = 1 \quad \text{and} \quad \frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1.$$ What is the value of $$\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}$$
  20. What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?

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MOlympiad: [Answer Keys] India Pre-Regional Mathematical Olympiad 2014
[Answer Keys] India Pre-Regional Mathematical Olympiad 2014
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