This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2015 Set B problems and solutions. You may find some solutions with hints too.
PRMO 2015 Set B, Problem 1:
A man walks a certain distance and rides back in $3 \frac{3}{4}$ hours; he could ride both ways in $2 \frac{1}{2}$ hours. How many hours would it take him to walk both ways?
PRMO 2015 Set B, Problem 2:
The equations $x^{2}-4 x+k=0$ and $x^{2}+k x-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$
PRMO 2015 Set B, Problem 3:
Positive integers $a$ and $b$ are such that $a+b=a / b+b / a$. What is the value of $a^{2}+b^{2} ?$
PRMO 2015 Set B, Problem 4:
How many line segments have both their endpoints located at the vertices of a given cube?
PRMO 2015 Set B, Problem 5:
Let $P(x)$ be a non-zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each positive integer $n,$ what is the value of $P(0) ?$
PRMO 2015 Set B, Problem 6:
How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square?
PRMO 2015 Set B, Problem 7:
Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(1243)=2+4=6$. What is the value of $E(1)+E(2)+E(3)+\cdots+E(100) ?$
PRMO 2015 Set B, Problem 8:
The figure below shows a broken piece of a circular plate made of glass.
$C$ is the midpoint of $A B,$ and $D$ is the midpoint of arc $A B .$ Given that $A B=24 \mathrm{~cm}$ and $C D=6 \mathrm{~cm},$ what is the radius of the plate in centimetres? (The figure is not drawn to scale.)
PRMO 2015 Set B, Problem 9:
What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12 ?$
PRMO 2015 Set B, Problem 10:
A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
PRMO 2015 Set B, Problem 11:
Let $a, b,$ and $c$ be real numbers such that $a-7 b+8 c=4$ and $8 a+4 b-c=7$. What is the value of $a^{2}-b^{2}+c^{2} ?$
PRMO 2015 Set B, Problem 12:
In rectangle $A B C D, A B=8$ and $B C=20 .$ Let $P$ be a point on $A D$ such that $\angle B P C=90^{\circ} .$ If $r_{1}, r_{2}, r_{3}$ are the radii of the incircles of triangles $A P B, B P C$ and $C P D,$ what is the value of $r_{1}+r_{2}+r_{3} ?$
PRMO 2015 Set B, Problem 13:
At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party?
PRMO 2015 Set B, Problem 14:
If $3^{x}+2^{y}=985$ and $3^{x}-2^{y}=473$, what is the value of $x y$ ?
PRMO 2015 Set B, Problem 15:
Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers, $x, y$ and $10 x+y,$ where $x$ and $y$ are digits. What is the sum of the digits of $n ?$
PRMO 2015 Set B, Problem 16:
In acute-angled triangle $A B C,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $B C .$ Let $F$ be the midpoint of $A C .$ Suppose $\angle B A E=40^{\circ} .$ If $\angle D A E=\angle D F E,$ what is the magnitude of $\angle A D F$ in degrees?
PRMO 2015 Set B, Problem 17:
Let $a, b$ and $c$ be such that $a+b+c=0$ and
$$
P=\frac{a^{2}}{2 a^{2}+b c}+\frac{b^{2}}{2 b^{2}+c a}+\frac{c^{2}}{2 c^{2}+a b}
$$
is defined. What is the value of $P ?$
PRMO 2015 Set B, Problem 18:
A subset $B$ of the set of first 100 positive integers has the property that no two elements of $B$ sum to $125 .$ What is the maximum possible number of elements in $B ?$
PRMO 2015 Set B, Problem 19:
The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37 ?$
PRMO 2015 Set B, Problem 20:
The circle $\omega$ touches the circle $\Omega$ internally at $P$. The centre $O$ of $\Omega$ is outside $\omega$. Let $X Y$ be a diameter of $\Omega$ which is also tangent to $\omega$. Assume $P Y>P X$. Let $P Y$ intersect $\omega$ at
Z. If $Y Z=2 P Z$, what is the magnitude of $\angle P Y X$ in degrees?
This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2015 Set B problems and solutions. You may find some solutions with hints too.
PRMO 2015 Set B, Problem 1:
A man walks a certain distance and rides back in $3 \frac{3}{4}$ hours; he could ride both ways in $2 \frac{1}{2}$ hours. How many hours would it take him to walk both ways?
PRMO 2015 Set B, Problem 2:
The equations $x^{2}-4 x+k=0$ and $x^{2}+k x-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$
PRMO 2015 Set B, Problem 3:
Positive integers $a$ and $b$ are such that $a+b=a / b+b / a$. What is the value of $a^{2}+b^{2} ?$
PRMO 2015 Set B, Problem 4:
How many line segments have both their endpoints located at the vertices of a given cube?
PRMO 2015 Set B, Problem 5:
Let $P(x)$ be a non-zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each positive integer $n,$ what is the value of $P(0) ?$
PRMO 2015 Set B, Problem 6:
How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square?
PRMO 2015 Set B, Problem 7:
Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(1243)=2+4=6$. What is the value of $E(1)+E(2)+E(3)+\cdots+E(100) ?$
PRMO 2015 Set B, Problem 8:
The figure below shows a broken piece of a circular plate made of glass.
$C$ is the midpoint of $A B,$ and $D$ is the midpoint of arc $A B .$ Given that $A B=24 \mathrm{~cm}$ and $C D=6 \mathrm{~cm},$ what is the radius of the plate in centimetres? (The figure is not drawn to scale.)
PRMO 2015 Set B, Problem 9:
What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12 ?$
PRMO 2015 Set B, Problem 10:
A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
PRMO 2015 Set B, Problem 11:
Let $a, b,$ and $c$ be real numbers such that $a-7 b+8 c=4$ and $8 a+4 b-c=7$. What is the value of $a^{2}-b^{2}+c^{2} ?$
PRMO 2015 Set B, Problem 12:
In rectangle $A B C D, A B=8$ and $B C=20 .$ Let $P$ be a point on $A D$ such that $\angle B P C=90^{\circ} .$ If $r_{1}, r_{2}, r_{3}$ are the radii of the incircles of triangles $A P B, B P C$ and $C P D,$ what is the value of $r_{1}+r_{2}+r_{3} ?$
PRMO 2015 Set B, Problem 13:
At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party?
PRMO 2015 Set B, Problem 14:
If $3^{x}+2^{y}=985$ and $3^{x}-2^{y}=473$, what is the value of $x y$ ?
PRMO 2015 Set B, Problem 15:
Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers, $x, y$ and $10 x+y,$ where $x$ and $y$ are digits. What is the sum of the digits of $n ?$
PRMO 2015 Set B, Problem 16:
In acute-angled triangle $A B C,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $B C .$ Let $F$ be the midpoint of $A C .$ Suppose $\angle B A E=40^{\circ} .$ If $\angle D A E=\angle D F E,$ what is the magnitude of $\angle A D F$ in degrees?
PRMO 2015 Set B, Problem 17:
Let $a, b$ and $c$ be such that $a+b+c=0$ and
$$
P=\frac{a^{2}}{2 a^{2}+b c}+\frac{b^{2}}{2 b^{2}+c a}+\frac{c^{2}}{2 c^{2}+a b}
$$
is defined. What is the value of $P ?$
PRMO 2015 Set B, Problem 18:
A subset $B$ of the set of first 100 positive integers has the property that no two elements of $B$ sum to $125 .$ What is the maximum possible number of elements in $B ?$
PRMO 2015 Set B, Problem 19:
The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37 ?$
PRMO 2015 Set B, Problem 20:
The circle $\omega$ touches the circle $\Omega$ internally at $P$. The centre $O$ of $\Omega$ is outside $\omega$. Let $X Y$ be a diameter of $\Omega$ which is also tangent to $\omega$. Assume $P Y>P X$. Let $P Y$ intersect $\omega$ at
Z. If $Y Z=2 P Z$, what is the magnitude of $\angle P Y X$ in degrees?
