[Answer Keys] India Pre-Regional Mathematical Olympiad 2018

1. A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$? 
2. In a quadrilateral $ABCD$, it is given that $AB = AD = 13$, $BC = CD = 20$, $BD = 24$. If $r$ is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to $r$? 
3. Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$. 
4. The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. 
5. Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$. 
6. Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ? 
7. A point $P$ in the interior of a regular hexagon is at distances $8,8,16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is radius of the circumscribed circle of the hexagon, what is the integer closest to $r$? 
8. Let $AB$ be a chord of a circle with centre $O$. Let $C$ be a point on the circle such that $\angle ABC =30^\circ$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^\circ$. Find the measure of $\angle CDO$ in degrees. 
9. Suppose $a$, $b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$? 
10. In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\dfrac{BC^2 + CA^2 + AB^2}{100}$. 
11. There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen? 
12. Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$. 
13. In a triangle $ABC$, right­ angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$. 
14. If $x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ$ and $y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \ldots \cos 86^\circ$, then what is the integer nearest to $\dfrac27 \log_2 \dfrac{y}{x}$ ? 
15. Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ? 
16. What is the value of $\displaystyle \sum_{1 \le i< j \le 10}(i+j)_{i+j=\text{odd}} - \sum_{1 \le i< j \le 10}(i+j)_{i+j=\text{even}} $ 
17. Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D$, $AB = DE = 17$, $BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$? 
18. If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ? 
19. Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$ 
20. Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$. 
21. Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1$, $G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC$, $\Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $? 
22. A positive integer $k$ is said to be good if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many good numbers are there? 
23. What is the largest positive integer $n$ such that $$\cfrac{a^2}{\cfrac{b}{29} + \cfrac{c}{31}}+\cfrac{b^2}{\cfrac{c}{29} + \cfrac{a}{31}}+\cfrac{c^2}{\cfrac{a}{29} + \cfrac{b}{31}} \ge n(a+b+c)$$ holds for all positive real numbers $a,b,c$. 
24. If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\dfrac{N}{100}$? 
25. Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ? 
26. What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common? 
27. What is the number of ways in which one can color the squares of a $4\times 4$ chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares? 
28. Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$. 
29. Let $D$ be an interior point of the side $BC$ of a triangle $ABC$. Let $I_1$ and $I_2$ be the incentres of triangles $ABD$ and $ACD$ respectively. Let $AI_1$ and $AI_2$ meet $BC$ in $E$ and $F$ respectively. If $\angle BI_1E = 60^\circ$, what is the measure of $\angle CI_2F$ in degrees? 
30. Let $P(x)$ = $a_0+a_1x+a_2x^2+\cdots +a_nx^n$ be a polynomial in which $a_i$ is non-negative integer for each $i \in$ {$0,1,2,3,....,n$} . If $P(1) = 4$ and $P(5) = 136$, what is the value of $P(3)$?

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