Part I : Solutions to Pre-RMO (Regional Mathematical Olympiad) Oct 2014

Question Paper Set A:

Qs 1) A natural number is such that $k^{2}<2014<(k+1)^{2}$. What is the largest prime factor of k?

Solution 1:

HInt: Think of perfect square numbers near 2014.

Hence, $44^{2}<2014<(44+1)^{2}$. And, the largest prime factor of 44 is 11. Hence, ans is 11.

Qs 2) The first term of a sequence is 2014. Each succeeding term is the sum of cubes of the digits of the previous term. What is the 2014th term of the sequence?

Solution 2:

Hint: Try the simplest solution…just keep calculating…:-)

$a_{1}=2014$

$a_{2}=2^{3}+1^{3}+4^{3}=73$

$a_{3}=7^{3}+3^{3}=343+27=370$

Clearly, $a_{2014}=370$, which is the answer.

Qs 3) Let ABCD be a convex quadrilateral with perpendicular diagonals. If AB=20, BC=70, and CD=90, then what is the value of DA?

Solution 3. let the diagonals meet at 0. Let OB=a,OC=b, OD=d and OA=c. Then, we can apply Pythagoras’s theorem to the 4 right angled triangles AOB, BOC, COD, DOA. We get

$a^{2}+b^{2}=70^{2}$

$b^{2}+d^{2}=90^{2}$

$d^{2}+c^{2}=DA^{2}$

$a^{2}+c^{2}=20^{2}$

Hence, we get, $a^{2}+b^{2}+c^{2}+d^{2}=90^{2}+20^{2}$. In the RHS, substitute for $a^{2}+b^{2}=70^{2}$ and you will get

$DA^{2}=c^{2}+d^{2}$.

Question 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?

Solution 4:$AB=c$;$AC=b$; $BC=a$; $b=3a$; $c=17$

Hint: use triangle inequalities:

$a+b>c$; $b+c>a$; $c+a>b$. Hence, $4a>17$ and

$a>17/4$.. Also, $a+17>3a$ and hence, $a<17/2$. So,

$17/4. So, we get $a=8$, $b=3a=24$

Ans. $8+24+17=49$.

Question 5. if real numbers a,b,c,d,e satisfy

$a+1=b+2=c+3=d+4=c+5=a+b+c+d+e+3$, what is the value of

$a^{2}+b^{2}+c^{3}+d^{4}+e^{5}$ ?

Solution 5. $b=a-1$, $c=b-1=a-2$, $d=c-1=a-3$,

$e=d-1=a-4$ so we get

$a+1=a+a-1+a-2+a-3+a-4+3$. Hence, $a=2$.

Ans. 10.

Question 6: What is the smallest possible natural number n for which the equation $x^{2}-ax+2014=0$ has integer roots?

Solution 6:

Hint: Use the discriminant formula.

$x=(n \pm \sqrt (n^{2}-8056))/2$.

So, just think of plugging in some values 🙂 and note that n has to be smallest.

Also, n has to be even and also $\sqrt (n^{2}-8056)$ has to be even and a perfect square root.

Ans. 91

Question 7: If $x^{(x)^{4}}=4$, what is the value of

$x^{(x)^{2}}+x^{(x)^{8}}$ ?

Solution 7:

Warning: Do not use logarithms as there is no formula for $log(M+N)$.

Hint: Try to input some numbers so that the original equation is satisfied.

$x=\sqrt (2)$.Ans.

Question 8: Let S be 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?

Solution: Use the definition of mean or average value followed by a little manipulation.

Ans. 4

Question 9: Natural numbers k,l,p,q are such that if a and b are roots of

$x^{2}-kx+l=0$, then $a+(1/b)$ and $b+(1/a)$ are the roots of $x^{2}-px+q=0$. What is the sum of all possible values of q?

Solution 9: Hint: Use the relationships between roots and coefficients:

$a+b=k$ — equation I

$ab=l$ — equation II

$a+(1/b)+b+(1/a)=p$ — equation III

$(a+1/b)(b+1/a)=q$ — equation IV

From equation IV, we get $ab+1+1+(1/ab)=q$, that is, $l+2+(1/l)=q$ But q is a natural number. Hence, $1/l$ is also a natural number and the only way this could be possible is $l=1$.

Hence, ans $q=4$.

Question 10 In a triangle ABC, X and Y are points on the segments AB and AC respectively, such that $AX:XB=1:2$ and $AY:YC=2:1$. If the area of the triangle AXY is 10, then what is the area of the triangle ABC?

Solution 10: Let $AX=k$, $BX=2k$, $AY=2m$, $CY=m$ where k and m are constants of proportionality.

Use: From trigonometry, $(1/2)bc \sin A = area of triangle ABC$ .

Here, $(1/2)k.2m.\sin A=10$ and hence, $mk \sin A=10$.

But, area of triangle ABC is $(1/2)(3k)(3m) \sin A=(9/2) \times 10=45$. Ans.

Question 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 $xy=x+y+(x,y)$?

To be discussed in the next blog.

Question 12:Let ABCD be a convex quadrilaterall with $\angle DAB = \angle BDC = 90 \deg$. Let the incircles of triangles ABD and BCD touch BD at P and Q, respectively, with P lying in between B and Q. If $AD=999$ and $PQ=200$, then what is the sum of the radii of the incircles of triangles ABD and BDC?

To be discussed in the next blog.

Question 13:For how many natural numbers n between 1 and 2014 (both inclusive) is $8n/(9999-n)$ an integer?

To be discussed in the next blog.

Question 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?

Solution 14: Key Idea: When two different fluids are mixed, they are in a certain proportion or ratio; this ratio of the two fluids remains the same when the mixture is poured out in different amounts.

Now, ratio of milk to coffee in Manjul’s mixture is $17/14$.

Let x be the constant of proportionality. Total mixture is 8 ounce. Hence,

$17x + 14x = 8$ and therefore, $x=8/31$. But, total milk is

$17x=(17/31) \times 8$ and total coffee is $14x = (14/31) \times 8$.

Hence, ans. 8.

Question 15: Let XOY be a triangle with $\angle XOY=90 \deg$. Let M and N be the midpoints of legs OX and OY, respectively. Suppose that $XN=19$ and $YM=22$. What is XY?\$

Solution 15. Let $XM=a=MO$ and $ON=b=NY$.

Apply Pythagoras’s theorem to right angled triangle MOY:

$a^{2}+(2b)^{2}=22 \times 22$.

Similarly, from right angled triangle XON, we get:

$(2a)^{2}+ b^{2}=19 \times 19$

Adding the above two equations, $5(a^{2}+b^{2})=845$ and hence,

$2(a^{2}+b^{2})=26$, which is the desired answer.

Question 16: In a triangle ABC, let I denote the incenter. Let the lines AI, BI, and CI intersect the incircle at P, Q and R, respectively. If $\angle BAC=40 \deg$, what is the value of $\angle QPR$ in degrees?

To be discussed in the next blog.

Question 17: For a natural number b, let $N(b)$ denote the number of natural numbers a for which the equation $x^{2}+ax+b=0$ has integer roots. What is the smallest value of b for which $N(b)=6$?

Question 18: Let f be a one-one function from the set of natural numbers to itself such that $f(mn)=f(m)f(n)$ for all natural numbers m and n. What is the least possible value of $f(999)$?

Question 19: Let $x_{1}, x_{2}, x_{3}, \ldots x_{2014}$ be real numbers different from 1, such that $x_{1}+x_{2}+x_{3}+ \ldots + x_{2014}=1$ and

$\frac {x_{1}}{1-x_{1}} + \frac {x_{2}}{1-x_{2}}+ \frac {x_{3}}{1-x_{3}}+ \ldots +\frac {x_{2014}}{1-x_{2014}}=1$.

What is the value of $\frac {x_{1}^{2}}{1-x_{1}}+\frac {x_{2}^{2}}{1-x_{2}}+\frac {x_{3}^{2}}{1-x_{3}}+\ldots + \frac {x_{2014}^{2}}{1-x_{2014}}$ ?

Solution.

$\frac {2x_{1}}{1-x_{1}}+\frac {2x_{2}}{1-x_{2}}+\frac {2x_{3}}{1-x_{3}}+\ldots+\frac {2x_{2014}}{1-x_{2014}}=2$

Let $S= \frac {x_{1}^{2}}{1-x_{1}}+\frac {x_{2}^{2}}{1-x_{2}}+\frac {x_{3}^{2}}{1-x_{3}}+\ldots+\frac {1-x_{2014}^{2}}{1-x_{2014}}$

hence, $S-2=\frac {x_{1}^{2}}{1-x_{1}}-\frac {2x_{1}}{1-x_{1}}+ \frac {x_{2}^{2}}{1-x_{2}}-\frac {2x_{2}}{1-x_{2}}+\ldots+\frac{x_{2014}^{2}}{1-x_{2014}}-\frac {2x_{2014}}{1-x_{2014}}$

But, $x_{1}+x_{2}+x_{3}+\ldots+x_{2014}=1$ and also given that

$\frac{x_{1}}{1-x_{1}}+\frac {x_{2}}{1-x{2}}+\frac {x_{3}}{1-x_{3}}+\ldots+\frac {x_{2014}}{1-x_{2014}}=1$

$(\frac {x_{1}}{1-x{1}}+1)+(\frac {x_{2}}{1-x{2}}+1)+(\frac {x_{3}}{1-x_{3}} +1)+\ldots +(\frac{x_{2014}}{1-x{2014}}+1)=2015$.

Hence, $S-2+2015=\frac {(1-x_{1})^{2}}{1-x_{1}} + \frac {(1-x_{2})^{2}}{1-x_{2}} +\ldots + \frac {(1-x_{2014})^{2}}{1-x_{2014}}$

which  equals $1-x_{1}+1-x_{2}+\ldots +1-x_{2014}=2014 -1=2013$.

Hence, $S=0$.

Question 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

$B \subseteq A$?

1. Abc