## Category Archives: Pre-RMO

### A nice dose of practice problems for IITJEE Foundation math and PreRMO

It is said that “practice makes man perfect”.

Problem 1:

Six boxes are numbered 1 through 6. How many ways are there to put 20 identical balls into  these boxes so that none of them is empty?

Problem 2:

How many ways are there to distribute n identical balls in m numbered boxes so that none of the boxes is empty?

Problem 3:

Six boxes are numbered 1 through 6. How many ways are there to distribute 20 identical balls between the boxes (this time some of the boxes can be empty)?

Finish this triad of problems now!

Nalin Pithwa.

### IITJEE Foundation Math and PRMO (preRMO) practice: another random collection of questions

Problem 1: Find the value of $\frac{x+2a}{2b--x} + \frac{x-2a}{2a+x} + \frac{4ab}{x^{2}-4b^{2}}$ when $x=\frac{ab}{a+b}$

Problem 2: Reduce the following fraction to its lowest terms:

$(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) \div (\frac{x+y+z}{x^{2}+y^{2}+z^{2}-xy-yz-zx} - \frac{1}{x+y+z})+1$

Problem 3: Simplify: $\sqrt[4]{97-56\sqrt{3}}$

Problem 4: If $a+b+c+d=2s$, prove that $4(ab+cd)^{2}-(a^{2}+b^{2}-c^{2}-d^{2})^{2}=16(s-a)(s-b)(s-c)(s-d)$

Problem 5: If a, b, c are in HP, show that $(\frac{3}{a} + \frac{3}{b} - \frac{2}{c})(\frac{3}{c} + \frac{3}{b} - \frac{2}{a})+\frac{9}{b^{2}}=\frac{25}{ac}$.

May u discover the joy of Math! 🙂 🙂 🙂

Nalin Pithwa.

### Pre-RMO (PRMO) Practice Problems

Pre-RMO days are back again. Here is a list of some of my random thoughts:

Problem 1:

There are five different teacups, three saucers, and four teaspoons in the “Tea Party” store. How many ways are there to buy two items with different names?

Problem 2:

We call a natural number “odd-looking” if all of its digits are odd. How many four-digit odd-looking numbers are there?

Problem 3:

We toss a coin three times. How many different sequences of heads and tails can we obtain?

Problem 4:

Each box in a 2 x 2 table can be coloured black or white. How many different colourings of the table are there?

Problem 5:

How many ways are there to fill in a Special Sport Lotto card? In this lotto, you must predict the results of 13 hockey games, indicating either a victory for one of two teams, or a draw.

Problem 6:

The Hermetian alphabet consists of only three letters: A, B and C. A word in this language is an arbitrary sequence of no more than four letters. How many words does the Hermetian language contain?

Problem 7:

A captain and a deputy captain must be elected in a soccer team with 11 players. How many ways are there to do this?

Problem 8:

How many ways are there to sew one three-coloured flag with three horizontal strips of equal height if we have pieces of fabric of six colours? We can distinguish the top of the flag from the bottom.

Problem 9:

How many ways are there to put one white and one black rook on a chessboard so that they do not attack each other?

Problem 10:

How many ways are there to put one white and one black king on a chessboard so that they do not attack each other?

I will post the answers in a couple of days.

Nalin Pithwa.

### Three in a row !!!

If my first were a 4,

And, my second were a 3,

What I am would be double,

The number you’d see.

For I’m only three digits,

Just three in a row,

So what must I be?

Don’t say you don’t know!

Cheers,

Nalin Pithwa.

### Cyclic Fractions for IITJEE foundation maths

Consider the expression

$\frac{1}{(a-b)(a-c)}+\frac{1}{(b-c)(b-a)}+\frac{1}{(c-a)(c-b)}$

Here, in finding the LCM of the denominators, it must be observed that there are not six different compound factors to be considered; for, three of them differ from the other three only in sign.

Thus,

$(a-c) = -(c-a)$

$(b-a) = -(a-b)$

$(c-b) = -(b-c)$

Hence, replacing the second factor in each denominator by its equivalent, we may write the expression in the form

$-\frac{1}{(a-b)(c-b)}-\frac{1}{(b-c)(a-b)}-\frac{1}{(c-a)(b-c)}$ call this expression 1

Now, the LCM is $(b-c)(c-a)(a-b)$

and the expression is $\frac{-(b-c)-(c-a)-(a-b)}{(b-c)(c-a)(a-b)}=0$.,

Some Remarks:

There is a peculiarity in the arrangement of this example, which is desirable to notice. In the expression 1, the letters occur in what is known as cyclic order; that is, b follows a, a follows c, c follows b. Thus, if a, b, c are arranged round the circumference of a circle, if we may start from any letter and move round in the direction of  the arrows, the other letters follow in cyclic  order; namely, abc, bca, cab.

The observance of this principle is especially important in a large class of examples in which the differences of three letters are involved. Thus, we are observing cyclic order when we write $b-c$, $c-a$, $a-b$, whereas we are violating order by the use of arrangements such as $b-c$, $a-c$, $a-b$, etc. It will always be found that the work is rendered shorter and easier by following cyclic order from the beginning, and adhering to it throughout the question.

Homework:

(1) Find the value of $\frac{a}{(a-b)(a-c)} + \frac{b}{(b-c)(b-a)} + \frac{c}{(c-a)(c-b)}$

2) Find the value of $\frac{b}{(a-b)(a-c)} + \frac{c}{(b-c)(b-a)} + \frac{a}{(c-a)(c-b)}$

3) Find the value of $\frac{z}{(x-y)(x-z)} + \frac{x}{(y-z)(y-x)} + \frac{y}{(z-x)(z-y)}$

4) Find the value of $\frac{y+z}{(x-y)(x-z)} + \frac{z+x}{(y-z)(y-x)} + \frac{x+y}{(z-x)(z-y)}$

5) Find the value of $\frac{b-c}{(a-b)(a-c)} + \frac{c-a}{(b-c)(b-a)} + \frac{a-b}{(c-a)(c-b)}$

More later,

Nalin Pithwa

### Geometry problems for Pre-RMO

Practice Quiz:

1. Prove that the median of a triangle which lies between two of its unequal sides forms a greater angle with the smaller of those sides.
2. Point A is given inside a triangle. Draw a line segment with end-points on the perimeter of the triangle so that the point divides the segment in half.
3. If the sides of a triangle are longer than 1000 inches, can its area be less than one inch?

Nalin Pithwa

### ISI or Pre-RMO practice problems — I

Problem #1.

A man started from home at 14:30 hours and drove to a village, arriving there when the clock indicated 15:15 hours. After staying for 25 min. he drove back by a different route of length $5/4$ times the first route at a rate twice as fast, reaching home at 16:00 hours. As compared to the clock at home, the village clock is

(a) 10 min slow

(b) 5 min slow

(c) 5 min fast

(d) 20 min fast

Problem #2.

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$, then

(a) $a=c$

(b) either $a=c$ or $a+b+c+d=0$

(c) $a+b+c+d=0$

(d) $a=c$ and $b=d$.

Problem #3.

In an election, 10% of the voters on the voters list did not cast their votes and 60 voters cast their ballot papers blank. There were only two candidates. The winner was supported by 47% of all voters in the list and he got 308 votes more than his rival. The number of voters on the list was

(A) 3600

(B) 6200

(C) 4575

(D) 6028

I hope to post more such questions every week,

Nalin Pithwa

### Pre-RMO — training

Problem 1:

If a, b, c, and d satisfy the equations

$a+7b+3c+5d=0$

$8a+4b+6c+2d=-16$

$2a+6b+4c+8d=16$

$5a+3b+7c+d=-16$

then what is the numerical value of $(a+d)(b+c)$?

Problem 2:

Suppose x and y are positive integers with $x>y$ and $3x+2y$ and $2x+3y$ when divided by 5, leave remainders 2 and 3, respectively. It follows that when $x-y$ is divided by 5, the remainder is necessarily equal to

(A) 2

(B) 1

(C) 4

(D) none of the foregoing numbers

Problem 3:

The number of different solutions $(x,y,z)$ of  the equation $x+y+z=10$, where each of x, y, and z is a positive integer is

(A) 36

(B) 121

(C) $10^{3}-10$

(D) $C_{3}^{10}-C_{2}^{10}$, which denote binomial coefficients

Problem 4:

The hands of a clock are observed simultaneously from 12.45 pm onwards. They will be observed to point in the same direction some time between

(A) 1:03 pm and 1:04 pm

(B) 1:04 pm and 1:05pm

(C) 1:05 pm and 1:06 pm

(D) 1:06 pm and 1:07 pm.

More later,

Nalin Pithwa

### Fun with Number Theory — Pre-RMO

Here is an elementary number theory problem which can be looked upon as practice problem for pre-RMO or even RMO or just plain fun with math.

Problem:

Find the least number whose last digit is 7 and which becomes 5 times larger when this last digit is carried to the beginning of the number.

Solution:

This is fun way to learn number theory or some Math. So, go ahead and try it. Your suggestions, answers, comments are welcome 🙂

More later,

Nalin Pithwa