This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2013 Set A problems and solutions. You may find some solutions with hints too.
There are 20 questions in the question paper and question carries 5 marks.
Time Duration: 2 hours
PRMO 2013 Set A, Problem 1:
What is the smallest positive integer $k$ such that $k\left(3^{3}+4^{3}+5^{3}\right)=a^{n}$ for some positive integers $a$ and $n,$ with $n>1 ?$
PRMO 2013 Set A, Problem 2:
Let $S_{n}=\sum_{k=0}^{n} \frac{1}{\sqrt{k+1}+\sqrt{k}} .$ What is the value of $\sum_{n=1}^{99} \frac{1}{S_{n}+S_{n-1}} ?$
PRMO 2013 Set A, Problem 3:
It is given that the equation $x^{2}+a x+20=0$ has integer roots. What is the sum of all possible values of $a ?$
PRMO 2013 Set A, Problem 4:
Three points $X, Y, Z$ are on a striaght line such that $X Y=10$ and $X Z=3$. What is the product of all possible values of $Y Z ?$
PRMO 2013 Set A, Problem 5:
There are $n-1$ red balls, $n$ green balls and $n+1$ blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is $299 .$ What is the value of $n ?$
PRMO 2013 Set A, Problem 6:
Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10 .$ Let $N$ be the smallest positive integer such that $S(N)=2013 .$ What is the value of $S(5 N+2013) ?$
PRMO 2013 Set A, Problem 7:
Let Akbar and Birbal together have $n$ marbles, where $n>0 .$
Akbar says to Birbal, " If I give you some marbles then you will have twice as many marbles as I will have." Birbal says to Akbar, " If I give you some marbles then you will have thrice as many marbles as I will have."
What is the minimum possible value of $n$ for which the above statements are true?
PRMO 2013 Set A, Problem 8:
Let $A D$ and $B C$ be the parallel sides of a trapezium $A B C D .$ Let $P$ and $Q$ be the midpoints of the diagonals $A C$ and $B D .$ If $A D=16$ and $B C=20,$ what is the length of $P Q ?$
PRMO 2013 Set A, Problem 9:
In a triangle $A B C,$ let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the noints $B . H . I . C$ lie on a circle, what is the magnitude of $\angle B O C$ in degrees?
PRMO 2013 Set A, Problem 10:
Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?
PRMO 2013 Set A, Problem 11:
Three real numbers $x, y, z$ are such that $x^{2}+6 y=-17, y^{2}+4 z=1$ and $z^{2}+2 x=2$. What is the value of $x^{2}+y^{2}+z^{2} ?$
PRMO 2013 Set A, Problem 12:
Let $A B C$ be an equilateral triangle. Let $P$ and $S$ be points on $A B$ and $A C$, respectively, and let $Q$ and $R$ be points on $B C$ such that $P Q R S$ is a rectangle. If $P Q=\sqrt{3} P S$ and the area of $P Q R S$ is $28 \sqrt{3}$, what is the length of $P C ?$
PRMO 2013 Set A, Problem 13:
To each element of the set $S={1,2, \ldots, 1000}$ a colour is assigned. Suppose that for any two elements $a . b$ of $S$. if 15 divides $a+b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
PRMO 2013 Set A, Problem 14:
Let $m$ be the smallest odd positive integer for which $1+2+\cdots+m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1+2+\cdots+n$ is a square of an integer. What is the value of $m+n ?$
PRMO 2013 Set A, Problem 15:
Let $A_{1}, B_{1}, C_{1}, D_{1}$ be the midpoints of the sides of a convex quadrilateral $A B C D$ and let A $_{2}$. $B_{2}$. $C_{2}, D_{2}$ be the midpoints of the sides of the quadrilateral $A_{1} B_{1} C_{1} D_{1}$. If $A_{2} B_{2} C_{2} D_{2}$ is a rectangle with sides 4 and $6,$ then what is the product of the lengths of the diagonals of $A B C D ?$
PRMO 2013 Set A, Problem 16:
Let $f(x)=x^{3}-3 x+b$ and $g(x)=x^{2}+b x-3,$ where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)=0$ and $g(x)=0$ have a common root?
PRMO 2013 Set A, Problem 17:
Let $S$ be a circle with centre $O$. A chord $A B,$ not a diameter, divides $S$ into two regions $R_{1}$ and $R_{2}$ such that $O$ belongs to $R_{2}$. Let $S_{1}$ be a circle with centre in $R_{1}$, touching $A B$ at $X$ and $S$ internally. Let $S_{2}$ be a circle with centre in $R_{2}$, touching $A B$ at $Y$, the circle $S$ internally and passing through the centre of $S .$ The point $X$ lies on the diameter passing through the centre of $S_{2}$ and $\angle Y X O=30^{\circ} .$ If the radius of $S_{2}$ is 100 then what is the radius of $S_{1} ?$
PRMO 2013 Set A, Problem 18:
What is the maximum possible value of $k$ for which 2013 can be written as a sum of $k$ consecutive positive integers?
PRMO 2013 Set A, Problem 19:
In a triangle $A B C$ with $\angle B C A=90^{\circ}$, the perpendicular bisector of $A B$ intersects segments $A B$ and $A C$ at $X$ and $Y$, respectively. If the ratio of the area of quadrilateral $B X Y C$ to the area of triangle $A B C$ is 13: 18 and $B C=12$ then what is the length of $A C ?$
PRMO 2013 Set A, Problem 20:
What is the sum (in base 10 ) of all the natural numbers less than 64 which have exactly three ones in their base 2 representation?
This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2013 Set A problems and solutions. You may find some solutions with hints too.
There are 20 questions in the question paper and question carries 5 marks.
Time Duration: 2 hours
PRMO 2013 Set A, Problem 1:
What is the smallest positive integer $k$ such that $k\left(3^{3}+4^{3}+5^{3}\right)=a^{n}$ for some positive integers $a$ and $n,$ with $n>1 ?$
PRMO 2013 Set A, Problem 2:
Let $S_{n}=\sum_{k=0}^{n} \frac{1}{\sqrt{k+1}+\sqrt{k}} .$ What is the value of $\sum_{n=1}^{99} \frac{1}{S_{n}+S_{n-1}} ?$
PRMO 2013 Set A, Problem 3:
It is given that the equation $x^{2}+a x+20=0$ has integer roots. What is the sum of all possible values of $a ?$
PRMO 2013 Set A, Problem 4:
Three points $X, Y, Z$ are on a striaght line such that $X Y=10$ and $X Z=3$. What is the product of all possible values of $Y Z ?$
PRMO 2013 Set A, Problem 5:
There are $n-1$ red balls, $n$ green balls and $n+1$ blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is $299 .$ What is the value of $n ?$
PRMO 2013 Set A, Problem 6:
Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10 .$ Let $N$ be the smallest positive integer such that $S(N)=2013 .$ What is the value of $S(5 N+2013) ?$
PRMO 2013 Set A, Problem 7:
Let Akbar and Birbal together have $n$ marbles, where $n>0 .$
Akbar says to Birbal, " If I give you some marbles then you will have twice as many marbles as I will have." Birbal says to Akbar, " If I give you some marbles then you will have thrice as many marbles as I will have."
What is the minimum possible value of $n$ for which the above statements are true?
PRMO 2013 Set A, Problem 8:
Let $A D$ and $B C$ be the parallel sides of a trapezium $A B C D .$ Let $P$ and $Q$ be the midpoints of the diagonals $A C$ and $B D .$ If $A D=16$ and $B C=20,$ what is the length of $P Q ?$
PRMO 2013 Set A, Problem 9:
In a triangle $A B C,$ let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the noints $B . H . I . C$ lie on a circle, what is the magnitude of $\angle B O C$ in degrees?
PRMO 2013 Set A, Problem 10:
Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?
PRMO 2013 Set A, Problem 11:
Three real numbers $x, y, z$ are such that $x^{2}+6 y=-17, y^{2}+4 z=1$ and $z^{2}+2 x=2$. What is the value of $x^{2}+y^{2}+z^{2} ?$
PRMO 2013 Set A, Problem 12:
Let $A B C$ be an equilateral triangle. Let $P$ and $S$ be points on $A B$ and $A C$, respectively, and let $Q$ and $R$ be points on $B C$ such that $P Q R S$ is a rectangle. If $P Q=\sqrt{3} P S$ and the area of $P Q R S$ is $28 \sqrt{3}$, what is the length of $P C ?$
PRMO 2013 Set A, Problem 13:
To each element of the set $S={1,2, \ldots, 1000}$ a colour is assigned. Suppose that for any two elements $a . b$ of $S$. if 15 divides $a+b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
PRMO 2013 Set A, Problem 14:
Let $m$ be the smallest odd positive integer for which $1+2+\cdots+m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1+2+\cdots+n$ is a square of an integer. What is the value of $m+n ?$
PRMO 2013 Set A, Problem 15:
Let $A_{1}, B_{1}, C_{1}, D_{1}$ be the midpoints of the sides of a convex quadrilateral $A B C D$ and let A $_{2}$. $B_{2}$. $C_{2}, D_{2}$ be the midpoints of the sides of the quadrilateral $A_{1} B_{1} C_{1} D_{1}$. If $A_{2} B_{2} C_{2} D_{2}$ is a rectangle with sides 4 and $6,$ then what is the product of the lengths of the diagonals of $A B C D ?$
PRMO 2013 Set A, Problem 16:
Let $f(x)=x^{3}-3 x+b$ and $g(x)=x^{2}+b x-3,$ where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)=0$ and $g(x)=0$ have a common root?
PRMO 2013 Set A, Problem 17:
Let $S$ be a circle with centre $O$. A chord $A B,$ not a diameter, divides $S$ into two regions $R_{1}$ and $R_{2}$ such that $O$ belongs to $R_{2}$. Let $S_{1}$ be a circle with centre in $R_{1}$, touching $A B$ at $X$ and $S$ internally. Let $S_{2}$ be a circle with centre in $R_{2}$, touching $A B$ at $Y$, the circle $S$ internally and passing through the centre of $S .$ The point $X$ lies on the diameter passing through the centre of $S_{2}$ and $\angle Y X O=30^{\circ} .$ If the radius of $S_{2}$ is 100 then what is the radius of $S_{1} ?$
PRMO 2013 Set A, Problem 18:
What is the maximum possible value of $k$ for which 2013 can be written as a sum of $k$ consecutive positive integers?
PRMO 2013 Set A, Problem 19:
In a triangle $A B C$ with $\angle B C A=90^{\circ}$, the perpendicular bisector of $A B$ intersects segments $A B$ and $A C$ at $X$ and $Y$, respectively. If the ratio of the area of quadrilateral $B X Y C$ to the area of triangle $A B C$ is 13: 18 and $B C=12$ then what is the length of $A C ?$
PRMO 2013 Set A, Problem 20:
What is the sum (in base 10 ) of all the natural numbers less than 64 which have exactly three ones in their base 2 representation?
