## Category Archives: Pre-RMO

### 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

### Percentage play

Alphonse bought two bicycles. He sold one to Bettany for 300 pounds making a loss of 25%, and one to Gemma also for 300 pounds making a profit of 25%. Overall, did he break even? If not, did he make a profit or loss, and by how much?

More later,

Nalin Pithwa

### Pre RMO type practice questions

1. Let $x_{1}, x_{2}, \ldots, x_{100}$ be positive integers such that $x_{i}+x_{i+1}=k$ for all i, where k is a constant. I $x_{10}=1$, find the value of $x_{1}$.
2. If $a_{0}=1, a_{1}=1$ and $a_{n}=a_{n-1}a_{n-2}+1$ for $n > 1$, then find out if $a_{465}$, $a_{466}$ are even or odd.
3. Two trains of equal length L, travelling at speeds $V_{1}$ and $V_{2}$ miles per hour in opposite directions, take T seconds to cross each other. Then, find L in feet (1 mile 1280 feet).
4. A salesman sold two pipes at Rs. 12 each. His profit on one was 20% and the loss on the other was 20%. Then, on the whole, what amount did he gain or lose or did he break even?
5. What is the digit in the units position of the integer $1! +2! +3! + \ldots +99!$?
6. Find the value of the following expression:

$(1+q)(1+q^{2})(1+q^{4})(1+q^{8})(1+q^{16})(1+q^{32})(1+q^{64})$ where $q \neq 1$.

Good luck for the ensuing oct pre RMO 🙂

Nalin Pithwa

### More Clock Problems

Example.

At what time between 4 and 5 o’clock will the minute-hand of a watch be 13 minutes in advance of the hour hand?

Solution.

Let x denote the required number of minutes after 4 o’clock; then, as the minute hand travels twelve times as fast as the hour hand, the hout hand will move over x/12 minute divisions in x minutes. At 4 o’clock, the minute hand is 20 divisions behind the hour hand, and the finally minute hand is 13 divisions in advance; therefore the minute hand moves over $20+13$, that is,, 33 divisions more than the hour hand.

Hence, $x=\frac{x}{12}+33$ which implies $\frac{11x}{12}=33$ and hence, $x=36$.

Thus, the time is 36 minutes past 4.

If the question be asked as follows: “At what times between 4 and 5 o’clock will there be 13 minutes between the two hands, then we must also take into consideration, the case when the minute hand is 13 divisions behind the hour hand. In this case, the minute hand gains $20-13$ or 7 divisions.

Hence,, $x=\frac{x}{12}+7$ which gives $x=7 \frac{7}{11}$

Therefore, the times are $7\frac{7}{11}$ past 4, and $36^{'}$ past 4.

Homework for fun:

1. At what time between one and two o’clock are the hands of a watch first at right angles?
2. At what time between 3 and 4 o’clock is the minute hand one minute ahead of the hour hand?
3. When are the hands of a clock together between the hours of 6 and 7?
4. It is between 2 and 3 o’clock, and in 10 minutes the minute hand will be as much before the hour hand as it is not behind it; what is the time?
5. At what times between 7 and 8 o’clock will the hands of a watch be at right angles to each other? When will they be in the same straight line?

Hope you had enough fun! 🙂

More fun later,

Nalin Pithwa