# PRMO Open Test

# Pre Regional Maths Olympiad, India

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- Question 1 of 30
##### 1. Question

A rectangle of 28 mtrs by 80 mtrs. A farmer with 2020 mtr fence uses it for internal fencing in congruent square plots. The sides of squares parallel to sides of figures. What is the largest number of square areas into which rectangular is fenced using all or part of 2020 mtrs.

CorrectIncorrect - Question 2 of 30
##### 2. Question

What is the row of pascal triangle where three cells one after the other comes in 7:8:9?

CorrectIncorrect - Question 3 of 30
##### 3. Question

Let x,y be natural numbers where \(x\sqrt{x}+y\sqrt{y}=183, x\sqrt{y}+y\sqrt{x}=182\), find \(\frac{9(x+y)}{5}\)

CorrectIncorrect - Question 4 of 30
##### 4. Question

What is number of natural numbers between 1 and 1000 that can be shown as difference of squares of two natural numbers?

CorrectIncorrect - Question 5 of 30
##### 5. Question

\(\Delta PQR\) has PQ=PR and PM=11 with point O on PM such that PO=10 and \(\angle QOR\)=3 \( \angle QPR\) with perimeter \(p +\sqrt{q}\). What is p+q?

CorrectIncorrect - Question 6 of 30
##### 6. Question

Five tents P,Q,R,S,T on a ground. Each tent is connected to another by tunnel. What is the number of ways a person starting from tent P returns to P after going to some tents once in every tent and taking same tunnel once.

CorrectIncorrect - Question 7 of 30
##### 7. Question

\(\Delta PQR\) \( \angle R\) is a right angle and \( \angle \)M is perpendicular to PQ. QM=\(29^{3}\) and cosQ=\(\frac{p}{q}\). What is p+q?

CorrectIncorrect - Question 8 of 30
##### 8. Question

Faces PQR and QRO of tetrahedron PQO meet at an angle 30. The area of the face PQR=120, the area of the face QRO =80, QR=10. What is the volume?

CorrectIncorrect - Question 9 of 30
##### 9. Question

Shaan multiplied a two digit number with a three digit number and did not give the multiplication sign where he kept 3 digit number to the right of 2 digit number a s a 5 digit number which is 9 times the product he is to obtain. What is the addition of 2 digit and 3 digit number?

CorrectIncorrect - Question 10 of 30
##### 10. Question

How many non-negative integer values we get from 5x+y+z+w=14?

CorrectIncorrect - Question 11 of 30
##### 11. Question

What is solution of t where p,q,s \(p>q>s\) natural numbers are in geometric progression in which \(p^{40}=q^{t}=s^{60}\)?

CorrectIncorrect - Question 12 of 30
##### 12. Question

In a collection of balls, the number of red balls and green balls are in ratio 4:3, on a certain game 8 red balls and 14 green balls were not present and that the number of red balls was the square of the number of green balls. find the total number of balls in the game.

CorrectIncorrect - Question 13 of 30
##### 13. Question

An equilateral triangle with points (0,0), (x,11), and (y,37), what is value of xy.

CorrectIncorrect - Question 14 of 30
##### 14. Question

Trapezoid OQRS has sides OQ=92, QR=50, RS=19, OS=70 with OQ parallel to RS. A circle with centre P on OQ is drawn tangent to QR and OS given that OP=\(\frac{a}{b}\) where a and b are relatively prime positive integers, find a+b.

CorrectIncorrect - Question 15 of 30
##### 15. Question

Triangle PQR has PQ=9 and QR:PR=40:41, what is greatest area of this triangle?

CorrectIncorrect - Question 16 of 30
##### 16. Question

Solve the equation to discover the least natural number solution to \(\cot{19x}=\frac{\cot{96}-1}{\cot{96}+1}\).All the angles are in degrees

CorrectIncorrect - Question 17 of 30
##### 17. Question

Let M be the sum of all numbers of the form \( \frac{p}{q}\) where p and q are divisors of 1200, find \([\frac{M}{15}]\)

CorrectIncorrect - Question 18 of 30
##### 18. Question

Ten balls are given on a table with no three balls lie on the same straight line. Five distinct sticks joining pairs of these balls are chosen at random, all sticks being equally likely. The probability that some four of the sticks form a quadrilateral whose vertices are among the ten given balls is \(\frac{p}{q}\) where p,q are coprime positive integers, find sum of digits of q-p.

CorrectIncorrect - Question 19 of 30
##### 19. Question

Point D is located inside right angled \(\Delta QRM\) so that triangles DQR,DRM,DMQ are all congruent. The sides of the triangles have lengths QR=13, RM=5, MQ=12 and the tangent of the angle DQR=\(\frac{m}{n}\) where m,n are relatively prime positive integers, find [tanx].

CorrectIncorrect - Question 20 of 30
##### 20. Question

Given that \(\sum_{n=1}^{35}cos5t=sin\frac{p}{q}\) where angles are measured in degrees, p,q are relatively prime positive integers that satisfy \(\frac{p}{q}<90\), find q.

CorrectIncorrect - Question 21 of 30
##### 21. Question

Let u,v,w,x be four positive numbers such that uvwx=1, u+\(\frac{1}{x}\)=7, v+\(\frac{1}{u}\)=11, w+\(\frac{1}{v}\)=13, x+\(\frac{1}{w}=\frac{p}{q}\) where p and q are coprime find \(|p-q|\).

CorrectIncorrect - Question 22 of 30
##### 22. Question

For complex numbers \(w_1,w_2,…,w_{202}\) polynomial \((x-w_1)^{10}(x-w_2)^{10}…(x-w_{202})^{10}\)=\(x^{2020}+20x^{2019}+19x^{2017}+f(x)\) such that f(x) is a polynomial with complex coefficients and with degree at most 2017. The values of \(|\sum_{1 \leq j \leq k \leq 202}w_jw_k|=\frac{p}{q}\), find\(|p-q|\).

CorrectIncorrect - Question 23 of 30
##### 23. Question

Let the solutions of \(a^{3}+5a^{2}+9a-11=0\) are x,y,z and solution of \(a^{3}+pa^{2}+qa+r=0\) are x+y, y+z, z+x, find q.

CorrectIncorrect - Question 24 of 30
##### 24. Question

Trapezoid OQRS has sides OQ=92, QR=50, RS=19, OS=70 with OQ parallel to RS. A circle with centre P on PQ is drawn tangent to QR and OS given that OP=\(\frac{a}{b}\) where a and b are relatively prime positive integers, find a+b.

CorrectIncorrect - Question 25 of 30
##### 25. Question

Let D be the product of the root of \((1+w^{2}+….+w^{8}+w^{10})=0\) that has a positive imaginary part with D=t(cosM+isinM) where \(0 \text{<} t\) and \(0 \leq M \leq 360\), find M.

CorrectIncorrect - Question 26 of 30
##### 26. Question

How many natural numbers less than or equal 9999 are there such that 11 divides the sum of the digits of each such number and 3 divides the actual number?

CorrectIncorrect - Question 27 of 30
##### 27. Question

Find the smallest integer \(t > 0\) such that however \((170)^{t}\) is expressed as the product of any two positive integers such that at least one of these two integers contain the digit 0.

CorrectIncorrect - Question 28 of 30
##### 28. Question

For each real number \(x\), let [x] denote the greatest integer that is less than or equal to x. For how many positive integers \(n\) is it true that \(n<5000\) and that \(\log_{3} n\) is a positive odd integer?

CorrectIncorrect - Question 29 of 30
##### 29. Question

What are the number of solutions of k in \(45^{45}\) which is gcd of \(15^{15},243^{243}\) and k.

CorrectIncorrect - Question 30 of 30
##### 30. Question

What are the number of solutions of k in \(45^{45}\) which is gcd of \(15^{15},243^{243}\) and k.

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