Vidyamanohar: 2012

Wednesday, March 28, 2012

Puzzle 43 - Locus of midpoints

Two circles ${C_1},\,{C_2}$ with different radii are given in the plane touching each other externally at $T$ . Consider points $A$ on ${C_1}$ and $B$ on ${C_1}$ , such that $AB$ subtends right angle at $T$ . Find the locus of midpoints of all such segments $AB$ .

Tuesday, March 27, 2012

Puzzle 42

Let $\inline f:R \to R$ is a function such that $\inline f(f(x)) = {x^2} - x + 1$ . Find the value of f(0).

Thursday, March 15, 2012

Puzzlee 41 - Polynomials + Trigonometry

If $\alpha < \beta < \gamma$ are three real roots of the equation ${x^3} - 3x - 1 = 0$ , show that ${\gamma ^2} - \gamma = {\beta ^2} - \alpha$ .

Wednesday, March 14, 2012

Puzzle 40 - Matrices

Let $A$ and $B$ are two different $3 \times 3$ matrices such that ${A^3} = {B^3}$ and ${A^2}B = {B^2}A$ . Does the matrix ${A^2} + {B^2}$ have inverse?

Tuesday, March 13, 2012

Puzzle 39 - 2012 Integration

Find the value of $\frac{{6539}}{{101}}\int\limits_{ - 1}^1 {\frac{{{t^{12}} + 31}}{{1 + {{2011}^t}}}dt}$

Thursday, March 08, 2012

Puzzle 38 - Point inside the Circle

Find the values of $\alpha$ for which the point $(\alpha - 1,\;\alpha + 1)$ lies in the larger segment of the circle ${x^2} + {y^2} - x - y - 6 = 0$ made by the chord whose equation is x + y – 2 = 0

Tuesday, March 06, 2012

Puzzle 37 - Extreme Values of Integral

Define the function $f(x) = \int\limits_0^{\frac{\pi }{2}} {\frac{{\cos |t - x|}}{{1 + \sin |t - x|}}} dt;\,0 \le x \le \pi$ , find the maximum and minimum values of f(x) in the interval $[0,\,\pi ]$

Saturday, February 25, 2012

Puzzle 36 - Functional Equation

Find the function f(x) if f(x) is a non constant and satisfies the equation $f(x) = {x^2} - \int\limits_0^1 {{{(f(t) + x)}^2}} dt$

Puzzle 35 - Complex system of equations

Find the complex roots of the system of equations

$(x + y)({x^2} - {y^2}) = 1176$

$(x - y)({x^2} + {y^2}) = 696$

Thursday, February 23, 2012

Puzzle 34 - Analytical Geometry

Vertex A of the triangle ABC is (14, 8). If the orthocenter of the triangle is H(8, 1) and the midpoint of the side BC is (3, -2), find the coordinates of the points B and C.

Puzzle 33 - Number Grid

In the grid shown, a number is to be placed in each smaller square so that the product of all the three numbers in any row, column, or diagonal is the same positive number. Find the sum of x and y.

Sunday, February 19, 2012

Puzzle 32 - Trigonometry

If $\alpha + \beta + \gamma = \pi$ and $\prod {\tan \left( {\frac{{\alpha + \beta - \gamma }}{4}} \right)} = 1$ find the value of $\sum {\sin \alpha } + \prod {\sin \alpha }$

Saturday, February 18, 2012

Puzzle 31 - Limit

Evaluate $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{\sqrt 2 }}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} } ...n\,terms} \right)$

Friday, February 17, 2012

Puzzle 30 - Parabola and Circle

The tangent and normal at the point $P(at^2, 2at)$ to the curve $y^2 = 4ax$ meet the x-axis and y-axis in T and G respectively, then find the angle at which the tangent at P to the curve is inclined to the tangent at P to the circle through the points T, P, G.

Tuesday, February 14, 2012

Puzzle 29 - Integration

If $\int {\frac{{1 - 7{{\cos }^2}x}}{{{{\sin }^7}x{{\cos }^2}x}}dx} = \frac{{f(x)}}{{{{\sin }^7}x}} + C$ , then find the value of $f'(0) + f''(\frac{\pi }{4})$

Monday, February 13, 2012

Puzzle 28 - Integration

Let ${\rm{f}}\left( {\rm{n}} \right)$ denotes the coefficient of ${{\rm{x}}^{{\rm{2}}0{\rm{12}}}}$ in the expansion of ${\left( {{\rm{1}} + {\rm{x}}} \right)^{\rm{n}}}$ . Then evaluate $\int\limits_0^1 {f( - t - 1)\left[ {\frac{1}{{t + 1}} + \frac{1}{{t + 2}} + \frac{1}{{t + 3}}...\frac{1}{{t + 2012}}} \right]} dt$

Puzzle 27 - Determinant

Define the function $f(x) = ({p_1} - x)({p_2} - x)({p_3} - x)...({p_n} - x);$ ${p_i} \in R$ . Let a and b are two unequal real numbers. Let M is an n×n matrix with all a’s above the diagonal and b’s below the diagonal and ${p_1},\,\,{p_2},\,\,{p_3},\,...{p_n}$ along the diagonal.
Show that the determinant of M is given by $\frac{{bf(a) - af(b)}}{{b - a}}$

Monday, January 23, 2012

Puzzle 26

Consider the three digit numbers 301, 431, 602, 715, 842, 856, 973, 986

They ALL satisfy the following properties:

1) All the digits are different

2) When the digits are placed along a cirlce, each digit is three times the previous digit, or one unit more, or two units more (while doing this, keep in mind that if this number comes out to be more than 9, only the units digit is taken). Hence 7 is followed by 1, 2 or 3. 3 is followed by 9, 0 or 1. If the number ends with the digit 8, then the first digit can be 4, 5 or 6.

Not only the above listed numbers, their cyclic permutations also satisfy the above properties.

Hence the above list gives 22 three digit numbers. We know, no number starts with a zero.

Now, try for four digited numbers that satisfy the above properties. How many such numbers are there, if we consider their cyclic permutations also as different number?

Sunday, January 22, 2012

Puzzle 25 - Remainders and quotients

Start with N, a six digit number. Subtract three from it, and then the new number formed should be divisible by 7. Take the six sevenths of this number. Call this new number i.e, $\frac{6}{7}(N - 3)$ as N1, and it should be divisible by 7. Continue this process till you form N6. Find N and N6. How long you can go like this?

Friday, January 13, 2012

Puzzle 24 Trigonometry

What is the minimum of the absolute value of the sum of all the six trigonometric functions?

Thursday, January 12, 2012

Puzzle 23 - Polynomials

If the equation $p{x^2} + (q + r)x + (s + t) = 0$ with real coefficients has both roots greater than 1, then show that the equation $p{x^4} + q{x^3} + r{x^2} + sx + t = 0$ has at least two real roots.

Saturday, January 07, 2012

Puzzle 22 - Cresents on sides of right triangle

ABC is a triangle right angled at C. Three semicircles are drawn on the sides AB, BC and CA as the respective diameters. If the sides a, b, c of the triangle are positive integers and the sum of the shaded areas is also to be a positive integer, how many distinct such triangles are there with perimeter at most 60 units?