Pre RMO Mock Test
By-div5252

Rules-
1. The answer to each question is an integer between 0 and 99 included.
2. Calculators are not allowed.
3. Time limit is 2:30 hours.
4. There are 30 questions and each question carries 3 marks. So the maximum score possible is 90.
5. Submit your answers to me through PM.
6. PM me if you think there is some mistake in question or are not able to understand it.

All the best!!!

1. Let $a,b,c$ be distinct non zero real numbers. If $a+\frac{1}{b}= b+\frac{1}{c}= c+\frac{1}{a}$ , then find the value of $a^2b^2c^2$ .

2. Let $f(x)=ax^2+bx+c$ where $a,b,c$ are integers. $f(1)=0$ , $40<f(6)<50$ and $60<f(7)<70$ . Then find $f(8)$ .

3. For any real $x$ the following holds-
$2(k-x)(x+\sqrt{x^2+k^2}) \le \alpha k^2$
Then find $\alpha^2$ .

4. For how many integers is the number $x^4-27x^2+26$ negative?

5. Let $r$ be the root of equation $x^2+2x+6$ . The value of
$(r+2)(r+3)(r+4)(r+5)=a$ . Find $a+200$ .

6. $f:R \mapsto R$
$f(x)+(x+\frac{1}{2})f(2-x)=1$ for all real $x$ .
Find $3f(0) +4f(2) +5f(1)$

7. Let $n$ be a natural number such that $n!+2$ is a perfect cube. The find the sum of values of all such $n$ .

8. Let $f(x)$ be a quadratic polynomial with $f(2)=14$ and $f(-2)=-6$ . The the coefficient of $x$ in $f(x)$ is

9. Let $x,y,z$ be positive reals such that
$\frac{1}{x}+ \frac{1}{y}+ \frac{1}{z}=5$
and $x+y+z=10$ .
Find the minimum value of $x^3+y^3+z^3-3$

10. Three real numbers $x,y,z$ are such that
$x^2+6y=-25$
$y^2+8z=-10$
$z^2+4x=6$
Find $xy+yz+zx$ .

11. A beetle sits on each square of $15 \times 15$ chessboard. At a signal each beetle crawls diagonally onto a neighbouring square. It may happen that several beetles sit on some square and none on others. Find the minimum possible number of free squares

12. Total number of solution for $x \in R$
$(4^x+2)(2-x)=6$ are

13. In quadrilateral $ABCD$ , $\angle DAB=100$ , $\angle ABC=115$ , $\angle ADC=105$ , $\angle DCB=40$ and $AC$ bisects $\angle DAB$ . Find $\angle DBC$ .

14. In $\triangle ABC$ , $AB=12$ , $AC=8$ , $BC=9$ . $D$ lies on $BC$ such that $BD:DC=5:4$ . Length of $AD$ is $a$ . Compute $\frac{9a^2}{4}-100$ .

15. A square with side length $x$ is inscribed in a right triangle with sides of length $7$ , $24$ , $25$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. Writing $\frac{x}{y}$ in lowest form compute the difference between the numerator and denominator in lowest form.

16. Let $f(x)=\frac{3}{9^x+3}$
Evaluate $\frac{1}{18} [f(\frac{1}{2017}) + f(\frac{2}{2017}) + …… +f(\frac{2016}{2017})]$ .

17. Find the sum of all $x$ which satisfy the equation $x^{x^2-6x+5}=1$

18. In $\triangle ABC$ , $AC=6$ , $AB=4$ . $AB$ is extended to $D$ such that $BD=5$ . Let $O$ be the circumcenter of $\triangle BDC$ . Find $\angle OBC$ if $\angle ACB=55$

19. On each face of cuboid is written the sum of area and perimeter. The three distinct sums are $281,341,433$ . Then find the sum of length, breadth and height of cuboid.

20. A basket has $3$ red and $7$ blue balls. Another basket has $7$ red and $N$ blue balls. A single ball is taken out randomly from each basket. The probability that both balls are of same colour is $0.56$ . Find $N$ .

21. Find the last two digits of $3^{2017}$ .

22. In $\triangle ABC$ , $AB=4$ , $AC=2$ , $BC=3$ . Bisector of $\angle A$ intersects $BC$ at $K$ . The straight line passing through $B$ and parallel to $AC$ intersects the extension of angle bisector $AK$ at point $M$ . $KM=a$ . Find $a^2$ .

23. Let $ABC$ be a triangle with $AB=16$ , $BC=20$ , $CA=21$ . Let $H$ be the orthocenter of $\triangle ABC$ . Find the distance b/w the circumcenters of triangles $AHB$ and $AHC$ .

24. In quadrilateral $ABCD$ , $\angle DAC=100$ . $\angle DBC=80$ , $\angle BCD=74$ and $BC=AD$ . Find $\angle ACD$ .

25. Find the rightmost non zero digit in expansion of $20(17!)$ .

26. Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. The probability that the two chosen segments have the same length is $\frac{x}{y}$ in simplest from. Find $x+y$ .

27. How many integers less than $2017$ but greater than $1000$ have the property that its unit digit is the sum of other digits. (Eg. $1023$ satisfies as $1+0+2=3$ ).

28. Let there be $N$ $6$ digit numbers formed using the digits $1,2,3$ only such that digit $1$ is not followed by digit $2$ . Find the sum of digits of $N$ .

29. The five digit number $17ab6$ is a perfect square. Find $ab+a+b$ .

30. Consider an analog clock. At time $T$ a mirror image of the clock shows a time $X$ a.m that is $5$ hours and $28$ minutes later than $T$ . Then the sum of hours and minutes in time $T$ is (Means if time $T$ is $10:15$ then sum is $10+15=25$ )