Saturday, December 13, 2025

2025/029) Prove that if $x^2+xy+y^2$ is divisible by 10 it is divisible by 100

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As 10 = 2 *5 let us check for both 2 and 5 We have $x^2+ xy+y^2 = x^2+ y(x+y)$ if x is odd the $x^2$ is odd and either y or x+y is even so t...
Friday, December 12, 2025

2025/028) How do you prove that $\cos20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ =\frac{1}{16}$

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Let $A= \cos20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ$ We have as 80 is double of 40 and 40 is double of 20 let us multiply both si...
Saturday, November 22, 2025

2025/027) Let a,b,c be the lengths of the sides of a triangle. Show that if $a^2+b^2+c^2=ab+bc+ca$ then the triangle is equilateral.

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 we have $a^2+b^2+c^2= ab + bc + ca$ Hence $a^2+b^2+c^2 -ab - bc -ca = 0$ Hence $2a62+2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0$ Or $(a^2-2ab + b^2)...
Wednesday, October 22, 2025

2025/026) prove $a(b-c)^2 +b(c-a)^2+c(a-b)^2+8abc=$ $(a+b) (b+c) (c+a)$

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 We have as on RHS (a+b) we can add 4abc to the 3rd term to get (a+b) as a factor and add 2abc to 1st and 2nd term that is distributing 8abc...
Friday, October 17, 2025

2025/025) How many numbers (n) are there between 1 and 200 such that $\frac{n}{2}$ , $\frac{n}{3}$ ,$\frac{2n+1}{5}$ are all composite natural numbers (CAT 2020)?

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First let us find the number such that $\frac{n}{2}$ ,  $\frac{n}{3}$ ,$\frac{2n+1}{5}$ are integers as  $\frac{n}{2}$ ,  $\frac{n}{3}$ are ...
Monday, October 13, 2025

2025/024) show that $3^{(2n + 1)}+ 2^{(n + 2)}$ is dvisible by 7

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we have   $3^{(2n + 1)}+ 2^{(n + 2)}$  $=3 *3^{2n}+ 4 *2^{n}$   $=3 *3^{2n}+ 4 *2^{n}$  $=3 *9^{n}+ 4 *2^{n}$    $=3 *2^{n}+ 4 *2^{n}$  as $...
Saturday, September 13, 2025

2025/023) Solve $\sqrt{(x+2)} > x $

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It is risky to square both sides as it shall give erroneous root. Let us first find the range of x as we can take the square root of a non -...