[Answer Keys] India Pre-Regional Mathematical Olympiad 2019

1. Consider the sequence of numbers $\left[n+\sqrt{2n}+\frac12\right]$, where $[x]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\ldots$ find $n_{12}$ 
2. If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$? 
3. Find the number of positive integers less than 101 that can not be written as the difference of two squares of integers. 
4. Let $a_1=24$ and form the sequence $a_n$, $n\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are $24,2534,253534,25353534,\ldots$. What is the least value of $n$ for which $a_n$ is divisible by $99$? 
5. Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$? 
6. Let $ABC$ be a triangle such that $AB=AC$. Suppose the tangent to the circumcircle of ABC at B is perpendicular to AC. Find angle ABC measured in degrees 
7. Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m)=20$ and $s(33m)=120$, what is the value of $s(3m)$? 
8. Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\dfrac{F_5(a,b)}{F_3(a,b)}$ an integer? 
9. The centre of the circle passing through the midpoints of the sides of am isosceles triangle $ABC$ lies on the circumcircle of triangle $ABC$. If the larger angle of triangle $ABC$ is $\alpha^{\circ}$ and the smaller one $\beta^{\circ}$ then what is the value of $\alpha-\beta$? 
10. One day I went for a walk in the morning at $x$ minutes past $5'O$ clock, where $x$ is a 2 digit number. When I returned, it was $y$ minutes past $6'O$ clock, and I noticed that I walked for exactly $x$ minutes and $y$ was a 2 digit number obtained by reversing the digits of $x$. How many minutes did I walk? 
11. Find the largest value of $a^b$ such that the positive integers $a,b>1$ satisfy $$a^bb^a+a^b+b^a=5329$$
12. Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $${10a+b:1\leq a\leq 5, 1\leq b\leq 5}$$where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$? 
13. Consider the sequence $$1,7,8,49,50,56,57,343\ldots$$which consists of sums of distinct powers of$7$, that is, $7^0$, $7^1$, $7^0+7^1$, $7^2$,$\ldots$ in increasing order. At what position will $16856$ occur in this sequence? 
14. Let $\mathcal{R}$ denote the circular region in the $xy-$plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $\mathcal{R}$ into four regions $\mathcal{R}_i ~ , ~i=1,2,3,4$. If $\mid \mathcal{R}_i \mid$ denotes the area of the region $\mathcal{R}_i$ and if $\mid \mathcal{R}_1 \mid >$ $\mid \mathcal{R}_2 \mid >$ $\mid \mathcal{R}_3 \mid > $ $\mid \mathcal{R}_4 \mid $, determine $\mid \mathcal{R}_1 \mid $ $-$ $\mid \mathcal{R}_2 \mid $ $-$ $\mid \mathcal{R}_3 \mid $ $+$ $\mid \mathcal{R}_4 \mid $. 
15. In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ? 
16. Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$. What is the value of $N$ ? 
17. Let $a,b,c$ be distinct positive integers such that $b+c-a$, $c+a-b$ and $a+b-c$ are all perfect squares. What is the largest possible value of $a+b+c$ smaller than $100$ ?
18. What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors ? 
19. If $15$ and $9$ are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer ? 
20. How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ? 
21. incorrect Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$ 
22. In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $$\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$$
23. Let $t$ be the area of a regular pentagon with each side equal to $1$. Let $P(x)=0$ be the polynomial equation with least degree, having integer coefficients, satisfied by $x=t$ and the $\gcd$ of all the coefficients equal to $1$. If $M$ is the sum of the absolute values of the coefficients of $P(x)$. What is the integer closest to $\sqrt{M}$ ? ($\sin 18^{\circ}=(\sqrt{5}-1)/2$) 
24. For $n \geq 1$, let $a_n$ be the number beginning with $n$ $9$'s followed by $744$; eg., $a_4=9999744$. Define $$f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \},$$ for $n\geq 1$. Find $f(1)+f(2)+f(3)+ \cdots + f(10)$. 
25. Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB$, $AC$ and $BD$ are integers and $AB-BD=3$, find $AC$. 
26. A friction-less board has the shape of an equilateral triangle of side length $1$ meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards the side $BC$. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form $\sqrt{N}$, where $N$ is an integer. What is the value of $N$ ? 
27. A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ?
28. In a triangle $ABC$, it is known that $\angle A=100^{\circ}$ and $AB=AC$. The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$ 
29. Let $ABC$ be an acute angled triangle with $AB=15$ and $BC=8$. Let $D$ be a point on $AB$ such that $BD=BC$. Consider points $E$ on $AC$ such that $\angle DEB=\angle BEC$. If $\alpha$ denotes the product of all possible values of $AE$, find $\lfloor \alpha \rfloor$ the integer part of $\alpha$. 
30. For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ $(\{x\}=x-\lfloor x \rfloor)$. Let $A$ denote the set of all real numbers $x$ satisfying $$\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}$$If $S$ is the sume of all numbers in $A$, find $\lfloor S \rfloor$

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