This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2014 problems and solutions. You may find some solutions with hints too.
PRMO 2014, Problem 1:
A natural number $k$ is such that $k^{2}<2014<(k+1)^{2}$. What is the largest prime factor of $k ?$
KEY CONCEPTS
ANSWER
11
Hint 1
Check the two integers $k , k+1$ such that $2014$ lies between $k^2$ and ${(k+1)}^2$
Hint 2
Find the square root of $2014$ and find the value of $k$ , $k+1$
Final Solution
$k^2 < 2014 <{(k+1)}^2$ gives the value of $k$ as $44$.
Hence the prime factorization of $44$ gives $2\times 2\times 11$ . Hence the largest prime factor of $k$ is $11$
PRMO 2014, Problem 2:
The first term of a sequence is $2014 .$ Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014^{\text {th }}$ term of the sequence?
Key Concepts
Hint 1
Find the first term second term third term 4th term and check for any pattern
Hint 2
First term is 2014
Second term is ${2^3}+{0^3}+{1^3}+{4^3}=73$
Third term is ${7^3}+{3^3}=370$
Fourth term is ${3^3}+{7^3}=370$
Final Answer
We find that from third term onwards each term is $370$. So the $ {2014}^{th}$ is $370$
PRMO 2014, Problem 3:
Let $A B C D$ be a convex quadrilateral with perpendicular diagonals. If $A B=20, B C=70$ and $C D=90,$ then what is the value of $D A ?$
Key Topics
Hint 1
Try to draw the diagram and apply Pythagoras theorem on each of the triangles.
Hint 2
In $\triangle AOB$ we have ${OA}^2+ {OB}^2=400......(i)$
In $\triangle BOC$ we have ${OB}^2+{OC}^2=4900.......(ii)$
In $\triangle COD$ we have ${OC}^2+{OD}^2=8100.....(iii)$
In $\triangle AOD$ we have ${OD}^2+{OA}^2={AD}^2.......(iv)$
Solution
From $(i) , (iii)$ we have ${OA}^2+{OB}^2+{OC}^2+{OD}^2=8500$
$({OA}^2+{OD}^2)+({OB}^2+{OC}^2=8500)$
${AD}^2+4900=8500$
Final Answer
${AD}^2=3600$ or $AD=60$
PRMO 2014, Problem 4:
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17 .$ What is the greatest possible perimeter of the triangle?
Key Concepts
Hint 1
The sum of two sides of a triangle is greater than the third side.
Hint 2
Assume the sides of a triangle to be of measurement $x$ and $3x$
Now $x+3x > 17 $ or $x > \frac{17}{4}$
$x+17> 3x$ give the equation $x<\frac{17}{2}$
Hint 3
$\frac{17}{4} < x <\frac{17}{2}$
Since $x$ is an integer hence $x = 5,6,7,8$
Solution
The maximum perimeter of the triangle is $8+24+17=49.$
PRMO 2014, Problem 5:
If real numbers $a, b, c, d, e$ satisfy
$$a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3$$
what is the value of $a^{2}+b^{2}+c^{2}+d^{2}+e^{2} ?$
PRMO 2014, Problem 6:
What is the smallest possible natural number $n$ for which the equation $x^{2}-n x+2014=0$ has integer roots?
Key Concepts
Hint 1
Find the product of the roots, and the sum of the roots.
Check the various combinations possible.
Find the least one among them.
Solution
Let the roots of the equation $x^2-nx+2014=0$ be $\alpha , \beta$
$\alpha \times \beta= 2014$
$\alpha+\beta=n$
Hence $2014 $ can be written as$(1\times 2014); (2\times 1007); (19\times 106); (38\times 53)$
Answer
The least possible value comes out to be $38+53= 91$
PRMO 2014, Problem 7:
If $x^{\left(x^{4}\right)}=4,$ what is the value of $x^{\left(x^{2}\right)}+x^{\left(x^{8}\right)} ?$
Key Concepts
Hint 1
Check the various possibilities for ${x}^4=4$
check and compare for each of these cases separately
Hint 2
${x}^{x^4} = 4$ gives ${x}^4=1$ or ${x}^4=2^2$ or ${x}^4={\sqrt 2}^4$
Hint 3
If $x^4=1$ then $x=1$ and also $x=4$. Not possible.
If $x^4=2$ then $x=\sqrt[4]{2}$ and $x=2$ not possible
If $x^4=4$ then $x= \sqrt[4]{4}=\sqrt{2}$ and $x=\sqrt{2}$
Solution
Thus $x=\sqrt {2}, {x}^2=2, {x}^8=16$
Now ${x}^{x^{2}}+{x}^{x^{8}}$ gives ${\sqrt{2}}^2+{\sqrt{2}}^{16}=258$
PRMO 2014, Problem 8:
Let. $S$ be a set of real numbers with mean $M$. If the means of the sets $S \cup{15}$ and $S \cup{15,1}$ are $M+2$ and $M+1$, respectively, then how many elements does $S$ have?
PRMO 2014, Problem 9:
Natural numbers $k, l, p$ and $q$ are such that if $a$ and $b$ are roots of $x^{2}-k x+l=0$ then $a+\frac{1}{b}$ and $b+\frac{1}{a}$ are the roots of $x^{2}-p x+q=0$. What is the sum of all possible values of $q$ ?
PRMO 2014, Problem 10:
In a triangle $A B C, X$ and $Y$ are points on the segments $A B$ and $A C,$ respectively, such that $A X: X B=1: 2$ and $A Y: Y C=2: 1$. If the area of triangle $A X Y$ is 10 then what is the area of triangle $A B \dot{C} ?$
PRMO 2014, Problem 11:
For natural numbers $x$ and $y,$ let $(x, y)$ denote the greatest common divisor of $x$ and $y .$ How many pairs of natural numbers $x$ and $y$ with $x \leq y$ satisfy the equation $x y=x+y+(x, y)$ ?
Discussion
PRMO 2014, Problem 12:
Let $A B C D$ be a convex quadrilateral with $\angle D A B=\angle B D C=90^{\circ}$. Let the incircles of triangles $A B D$ and $B C D$ touch $B D$ at $P$ and $Q,$ respectively, with $P$ lying in between $B$ and $Q .$ If $A D=999$ and $P Q=200$ then what is the sum of the radii of the incircles of triangles $A B D$ and $B D C ?$
PRMO 2014, Problem 13:
For how many ratural numbers $n$ between 1 and 2014 (both inclusive) is $\frac{8 n}{9999-n}$ an integer?
PRMO 2014, Problem 14:
One morning, each member of Manjul's family drank an 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1 / 7$ -th of the total amount of milk and $2 / 17$ -th of the total amount of coffee. How many people are there in Manjul's family?
PRMO 2014, Problem 15:
Let $X O Y$ be a triangle with $\angle X O Y=90^{\circ} .$ Let $M$ and $N$ be the midpoints of legs $O X$ and OY, respectively. Suppose that $X N=19$ and $Y M=22 .$ What is $X Y ?$
PRMO 2014, Problem 16:
In a triangle $A B C,$ let $I$ denote the incenter. Let the lines $A I, B I$ and $C I$ intersect the incircle at $P, Q$ and $R$, respectively. If $\angle B A C=40^{\circ}$, what is the value of $\angle Q P R$ in degrees?
PRMO 2014, Problem 17:
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^{2}+a x+b=0$ has integer roots. What is the smallest value of $b$ for which $N(b)=20 ?$
PRMO 2014, Problem 18:
Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(m n)=$ $f(m) f(n)$ for all natural numbers $m$ and $n .$ What is the least possible value of $f(999) ?$
PRMO 2014, Problem 19:
Let $x_{1}, x_{2}, \cdots, x_{2014}$ be real numbers different from $1,$ such that $x_{1}+x_{2}+\cdots+x_{2014}=1$ and
$$
\frac{x_{1}}{1-x_{1}}+\frac{x_{2}}{1-x_{2}}+\cdots+\frac{x_{2014}}{1-x_{2014}}=1
$$
What is the value of
$$
\frac{x_{1}^{2}}{1-x_{1}}+\frac{x_{2}^{2}}{1-x_{2}}+\frac{x_{3}^{2}}{1-x_{3}}+\cdots+\frac{x_{2014}^{2}}{1-x_{2014}} ?
$$
PRMO 2014, Problem 20:
What is the number of ordered pairs $(A, B)$ where $A$ and $B$ are subsets of ${1,2, \ldots, 5}$ such that neither $A \subseteq B$ nor $B \subseteq A ?$
This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2014 problems and solutions. You may find some solutions with hints too.
PRMO 2014, Problem 1:
A natural number $k$ is such that $k^{2}<2014<(k+1)^{2}$. What is the largest prime factor of $k ?$
KEY CONCEPTS
ANSWER
11
Hint 1
Check the two integers $k , k+1$ such that $2014$ lies between $k^2$ and ${(k+1)}^2$
Hint 2
Find the square root of $2014$ and find the value of $k$ , $k+1$
Final Solution
$k^2 < 2014 <{(k+1)}^2$ gives the value of $k$ as $44$.
Hence the prime factorization of $44$ gives $2\times 2\times 11$ . Hence the largest prime factor of $k$ is $11$
PRMO 2014, Problem 2:
The first term of a sequence is $2014 .$ Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014^{\text {th }}$ term of the sequence?
Key Concepts
Hint 1
Find the first term second term third term 4th term and check for any pattern
Hint 2
First term is 2014
Second term is ${2^3}+{0^3}+{1^3}+{4^3}=73$
Third term is ${7^3}+{3^3}=370$
Fourth term is ${3^3}+{7^3}=370$
Final Answer
We find that from third term onwards each term is $370$. So the $ {2014}^{th}$ is $370$
PRMO 2014, Problem 3:
Let $A B C D$ be a convex quadrilateral with perpendicular diagonals. If $A B=20, B C=70$ and $C D=90,$ then what is the value of $D A ?$
Key Topics
Hint 1
Try to draw the diagram and apply Pythagoras theorem on each of the triangles.
Hint 2
In $\triangle AOB$ we have ${OA}^2+ {OB}^2=400......(i)$
In $\triangle BOC$ we have ${OB}^2+{OC}^2=4900.......(ii)$
In $\triangle COD$ we have ${OC}^2+{OD}^2=8100.....(iii)$
In $\triangle AOD$ we have ${OD}^2+{OA}^2={AD}^2.......(iv)$
Solution
From $(i) , (iii)$ we have ${OA}^2+{OB}^2+{OC}^2+{OD}^2=8500$
$({OA}^2+{OD}^2)+({OB}^2+{OC}^2=8500)$
${AD}^2+4900=8500$
Final Answer
${AD}^2=3600$ or $AD=60$
PRMO 2014, Problem 4:
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17 .$ What is the greatest possible perimeter of the triangle?
Key Concepts
Hint 1
The sum of two sides of a triangle is greater than the third side.
Hint 2
Assume the sides of a triangle to be of measurement $x$ and $3x$
Now $x+3x > 17 $ or $x > \frac{17}{4}$
$x+17> 3x$ give the equation $x<\frac{17}{2}$
Hint 3
$\frac{17}{4} < x <\frac{17}{2}$
Since $x$ is an integer hence $x = 5,6,7,8$
Solution
The maximum perimeter of the triangle is $8+24+17=49.$
PRMO 2014, Problem 5:
If real numbers $a, b, c, d, e$ satisfy
$$a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3$$
what is the value of $a^{2}+b^{2}+c^{2}+d^{2}+e^{2} ?$
PRMO 2014, Problem 6:
What is the smallest possible natural number $n$ for which the equation $x^{2}-n x+2014=0$ has integer roots?
Key Concepts
Hint 1
Find the product of the roots, and the sum of the roots.
Check the various combinations possible.
Find the least one among them.
Solution
Let the roots of the equation $x^2-nx+2014=0$ be $\alpha , \beta$
$\alpha \times \beta= 2014$
$\alpha+\beta=n$
Hence $2014 $ can be written as$(1\times 2014); (2\times 1007); (19\times 106); (38\times 53)$
Answer
The least possible value comes out to be $38+53= 91$
PRMO 2014, Problem 7:
If $x^{\left(x^{4}\right)}=4,$ what is the value of $x^{\left(x^{2}\right)}+x^{\left(x^{8}\right)} ?$
Key Concepts
Hint 1
Check the various possibilities for ${x}^4=4$
check and compare for each of these cases separately
Hint 2
${x}^{x^4} = 4$ gives ${x}^4=1$ or ${x}^4=2^2$ or ${x}^4={\sqrt 2}^4$
Hint 3
If $x^4=1$ then $x=1$ and also $x=4$. Not possible.
If $x^4=2$ then $x=\sqrt[4]{2}$ and $x=2$ not possible
If $x^4=4$ then $x= \sqrt[4]{4}=\sqrt{2}$ and $x=\sqrt{2}$
Solution
Thus $x=\sqrt {2}, {x}^2=2, {x}^8=16$
Now ${x}^{x^{2}}+{x}^{x^{8}}$ gives ${\sqrt{2}}^2+{\sqrt{2}}^{16}=258$
PRMO 2014, Problem 8:
Let. $S$ be a set of real numbers with mean $M$. If the means of the sets $S \cup{15}$ and $S \cup{15,1}$ are $M+2$ and $M+1$, respectively, then how many elements does $S$ have?
PRMO 2014, Problem 9:
Natural numbers $k, l, p$ and $q$ are such that if $a$ and $b$ are roots of $x^{2}-k x+l=0$ then $a+\frac{1}{b}$ and $b+\frac{1}{a}$ are the roots of $x^{2}-p x+q=0$. What is the sum of all possible values of $q$ ?
PRMO 2014, Problem 10:
In a triangle $A B C, X$ and $Y$ are points on the segments $A B$ and $A C,$ respectively, such that $A X: X B=1: 2$ and $A Y: Y C=2: 1$. If the area of triangle $A X Y$ is 10 then what is the area of triangle $A B \dot{C} ?$
PRMO 2014, Problem 11:
For natural numbers $x$ and $y,$ let $(x, y)$ denote the greatest common divisor of $x$ and $y .$ How many pairs of natural numbers $x$ and $y$ with $x \leq y$ satisfy the equation $x y=x+y+(x, y)$ ?
Discussion
PRMO 2014, Problem 12:
Let $A B C D$ be a convex quadrilateral with $\angle D A B=\angle B D C=90^{\circ}$. Let the incircles of triangles $A B D$ and $B C D$ touch $B D$ at $P$ and $Q,$ respectively, with $P$ lying in between $B$ and $Q .$ If $A D=999$ and $P Q=200$ then what is the sum of the radii of the incircles of triangles $A B D$ and $B D C ?$
PRMO 2014, Problem 13:
For how many ratural numbers $n$ between 1 and 2014 (both inclusive) is $\frac{8 n}{9999-n}$ an integer?
PRMO 2014, Problem 14:
One morning, each member of Manjul's family drank an 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1 / 7$ -th of the total amount of milk and $2 / 17$ -th of the total amount of coffee. How many people are there in Manjul's family?
PRMO 2014, Problem 15:
Let $X O Y$ be a triangle with $\angle X O Y=90^{\circ} .$ Let $M$ and $N$ be the midpoints of legs $O X$ and OY, respectively. Suppose that $X N=19$ and $Y M=22 .$ What is $X Y ?$
PRMO 2014, Problem 16:
In a triangle $A B C,$ let $I$ denote the incenter. Let the lines $A I, B I$ and $C I$ intersect the incircle at $P, Q$ and $R$, respectively. If $\angle B A C=40^{\circ}$, what is the value of $\angle Q P R$ in degrees?
PRMO 2014, Problem 17:
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^{2}+a x+b=0$ has integer roots. What is the smallest value of $b$ for which $N(b)=20 ?$
PRMO 2014, Problem 18:
Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(m n)=$ $f(m) f(n)$ for all natural numbers $m$ and $n .$ What is the least possible value of $f(999) ?$
PRMO 2014, Problem 19:
Let $x_{1}, x_{2}, \cdots, x_{2014}$ be real numbers different from $1,$ such that $x_{1}+x_{2}+\cdots+x_{2014}=1$ and
$$
\frac{x_{1}}{1-x_{1}}+\frac{x_{2}}{1-x_{2}}+\cdots+\frac{x_{2014}}{1-x_{2014}}=1
$$
What is the value of
$$
\frac{x_{1}^{2}}{1-x_{1}}+\frac{x_{2}^{2}}{1-x_{2}}+\frac{x_{3}^{2}}{1-x_{3}}+\cdots+\frac{x_{2014}^{2}}{1-x_{2014}} ?
$$
PRMO 2014, Problem 20:
What is the number of ordered pairs $(A, B)$ where $A$ and $B$ are subsets of ${1,2, \ldots, 5}$ such that neither $A \subseteq B$ nor $B \subseteq A ?$
