This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2012 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 2012 Set A, Problem 1:
Rama was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and divided the result by 3. She got 43 as the answer. What would have been her answer if she had solved the problem correctly?
PRMO 2012 Set A, Problem 2:
A triangle with perimeter 7 has integer side lengths. What is the maximum nossible ares of such a triangle?
PRMO 2012 Set A, Problem 3:
For how many pairs of positive integers $(x, y)$ is $x+3 y=100 ?$
PRMO 2012 Set A, Problem 4:
The letters $R, M,$ and $O$ represent whole numbers. If $R \times M \times O=240, R \times O+M=46$ and $R+M \times O=64,$ what is the value of $R+M+O ?$
PRMO 2012 Set A, Problem 5:
Let $S_{n}=n^{2}+20 n+12, n$ a positive integer. What is the sum of all possible values of $n$ for which $S_{n}$ is a perfect square?
PRMO 2012 Set A, Problem 6:
A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?
PRMO 2012 Set A, Problem 7:
In $\triangle A B C,$ we have $A C=B C=7$ and $A B=2$. Suppose that $D$ is a point on line $A B$ such that $B$ lies between $A$ and $D$ and $C D=8 .$ What is the length of the segment $B D ?$
PRMO 2012 Set A, Problem 8:
In rectangle $A B C D, A B=5$ and $B C=3$. Points $F$ and $G$ are on line segment $C D$ so that $D F=1$ and $G C=2$. Lines $A F$ and $B G$ intersect. at. $E$. What is the area of $\triangle A E B ?$
PRMO 2012 Set A, Problem 9:
Suppose that $4^{X_{1}}=5,5^{X_{2}}=6,6^{X_{3}}=7, \ldots, 126^{X_{123}}=127,127^{X_{124}}=128 .$ What is the
value of the product $X_{1} X_{2} \ldots X_{124} ?$
PRMO 2012 Set A, Problem 10:
$A B C D$ is a square and $A B=1 .$ Equilateral triangles $A Y B$ and $C X D$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $X Y ?$
PRMO 2012 Set A, Problem 11:
Let $P(n)=(n+1)(n+3)(n+5)(n+7)(n+9) .$ What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n ?$
PRMO 2012 Set A, Problem 12:
If $\frac{1}{\sqrt{2011+\sqrt{2011^{2}-1}}}=\sqrt{m}-\sqrt{n},$ where $m$ and $n$ are positive integers, what is the value of $m+n ?$
PRMO 2012 Set A, Problem 13:
If $a=b-c, b=c-d, c=d-a$ and $a b c d \neq 0$ then what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} ?$
PRMO 2012 Set A, Problem 14:
$O$ and $I$ are the circumcentre and incentre of $\triangle A B C$ respectively. Suppose $O$ lies in the interior of $\triangle A B C$ and $I$ lies on the circle passing through $B, O,$ and $C .$ What is the magnitude of $\angle B A C$ in degrees?
PRMO 2012 Set A, Problem 15:
How many non-negative integral values of $x$ satisfy the equation $\left[\frac{x}{5}\right]=\left[\frac{x}{7}\right] ?$
(Here $[x]$ denotes the greatest integer less than or equal to $x$. For example $[3.4]=3$ and $[-2.3]=-3 .)$
PRMO 2012 Set A, Problem 16:
Let $N$ be the set of natural numbers. Suppose $f: N \rightarrow N$ is a function satisfying the following conditions.
(a) $f(m n)=f(m) f(n)$
(b) $f(m)<f(n)$ if $m<n$;
(c) $f(2)=2$.
What is the value of $\sum_{k=1}^{20} f(k) ?$
PRMO 2012 Set A, Problem 17:
Let. $x_{1}$. $x_{2}, x_{3}$ be the roots of the equation $x^{3}+3 x+5=0$. What is the value of the expression
$$
\left(x_{1}+\frac{1}{x_{1}}\right)\left(x_{2}+\frac{1}{x_{2}}\right)\left(x_{3}+\frac{1}{x_{3}}\right) ?
$$
PRMO 2012 Set A, Problem 18:
What is the sum of the squares of the roots of the equation $x^{2}-7[x]+5=0 ?$ (Here $[x]$ denotes the greatest integer less than or equal to $x$. For example $[3.4]=3$ and $[-2.3]=-3 .)$
PRMO 2012 Set A, Problem 19:
How many integer pairs $(x, y)$ satisfy $x^{2}+4 y^{2}-2 x y-2 x-4 y-8=0 ?$
PRMO 2012 Set A, Problem 20:
PS is a line segment of length 4 and $O$ is the midpoint of $P S$. A semicircular arc is drawn with $P S$ as diameter. Let $X$ be the midpoint of this arc. $Q$ and $R$ are points on the arc $P X S$ such that $Q R$, is parallel to $P S$ and the semicircular arc drawn with $Q R$ as diameter PA such by the semicircular arcs?
Solution :
This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2012 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 2012 Set A, Problem 1:
Rama was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and divided the result by 3. She got 43 as the answer. What would have been her answer if she had solved the problem correctly?
PRMO 2012 Set A, Problem 2:
A triangle with perimeter 7 has integer side lengths. What is the maximum nossible ares of such a triangle?
PRMO 2012 Set A, Problem 3:
For how many pairs of positive integers $(x, y)$ is $x+3 y=100 ?$
PRMO 2012 Set A, Problem 4:
The letters $R, M,$ and $O$ represent whole numbers. If $R \times M \times O=240, R \times O+M=46$ and $R+M \times O=64,$ what is the value of $R+M+O ?$
PRMO 2012 Set A, Problem 5:
Let $S_{n}=n^{2}+20 n+12, n$ a positive integer. What is the sum of all possible values of $n$ for which $S_{n}$ is a perfect square?
PRMO 2012 Set A, Problem 6:
A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?
PRMO 2012 Set A, Problem 7:
In $\triangle A B C,$ we have $A C=B C=7$ and $A B=2$. Suppose that $D$ is a point on line $A B$ such that $B$ lies between $A$ and $D$ and $C D=8 .$ What is the length of the segment $B D ?$
PRMO 2012 Set A, Problem 8:
In rectangle $A B C D, A B=5$ and $B C=3$. Points $F$ and $G$ are on line segment $C D$ so that $D F=1$ and $G C=2$. Lines $A F$ and $B G$ intersect. at. $E$. What is the area of $\triangle A E B ?$
PRMO 2012 Set A, Problem 9:
Suppose that $4^{X_{1}}=5,5^{X_{2}}=6,6^{X_{3}}=7, \ldots, 126^{X_{123}}=127,127^{X_{124}}=128 .$ What is the
value of the product $X_{1} X_{2} \ldots X_{124} ?$
PRMO 2012 Set A, Problem 10:
$A B C D$ is a square and $A B=1 .$ Equilateral triangles $A Y B$ and $C X D$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $X Y ?$
PRMO 2012 Set A, Problem 11:
Let $P(n)=(n+1)(n+3)(n+5)(n+7)(n+9) .$ What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n ?$
PRMO 2012 Set A, Problem 12:
If $\frac{1}{\sqrt{2011+\sqrt{2011^{2}-1}}}=\sqrt{m}-\sqrt{n},$ where $m$ and $n$ are positive integers, what is the value of $m+n ?$
PRMO 2012 Set A, Problem 13:
If $a=b-c, b=c-d, c=d-a$ and $a b c d \neq 0$ then what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} ?$
PRMO 2012 Set A, Problem 14:
$O$ and $I$ are the circumcentre and incentre of $\triangle A B C$ respectively. Suppose $O$ lies in the interior of $\triangle A B C$ and $I$ lies on the circle passing through $B, O,$ and $C .$ What is the magnitude of $\angle B A C$ in degrees?
PRMO 2012 Set A, Problem 15:
How many non-negative integral values of $x$ satisfy the equation $\left[\frac{x}{5}\right]=\left[\frac{x}{7}\right] ?$
(Here $[x]$ denotes the greatest integer less than or equal to $x$. For example $[3.4]=3$ and $[-2.3]=-3 .)$
PRMO 2012 Set A, Problem 16:
Let $N$ be the set of natural numbers. Suppose $f: N \rightarrow N$ is a function satisfying the following conditions.
(a) $f(m n)=f(m) f(n)$
(b) $f(m)<f(n)$ if $m<n$;
(c) $f(2)=2$.
What is the value of $\sum_{k=1}^{20} f(k) ?$
PRMO 2012 Set A, Problem 17:
Let. $x_{1}$. $x_{2}, x_{3}$ be the roots of the equation $x^{3}+3 x+5=0$. What is the value of the expression
$$
\left(x_{1}+\frac{1}{x_{1}}\right)\left(x_{2}+\frac{1}{x_{2}}\right)\left(x_{3}+\frac{1}{x_{3}}\right) ?
$$
PRMO 2012 Set A, Problem 18:
What is the sum of the squares of the roots of the equation $x^{2}-7[x]+5=0 ?$ (Here $[x]$ denotes the greatest integer less than or equal to $x$. For example $[3.4]=3$ and $[-2.3]=-3 .)$
PRMO 2012 Set A, Problem 19:
How many integer pairs $(x, y)$ satisfy $x^{2}+4 y^{2}-2 x y-2 x-4 y-8=0 ?$
PRMO 2012 Set A, Problem 20:
PS is a line segment of length 4 and $O$ is the midpoint of $P S$. A semicircular arc is drawn with $P S$ as diameter. Let $X$ be the midpoint of this arc. $Q$ and $R$ are points on the arc $P X S$ such that $Q R$, is parallel to $P S$ and the semicircular arc drawn with $Q R$ as diameter PA such by the semicircular arcs?
Solution :
