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Cauchy Schwarz Problem: Let be a polynomial with non-negative coefficients.Prove that if for ,then the same inequality holds for each . Discussion: Cauchy Schwarz's Inequality: Suppose for real numbers (\ a_{i},b_{i}), where (\ i\in{1,2,\dots,n}) we can say that $${\sum_{i=1}^{n}a_{i}^2}{\sum_{i=1}^{n}b_{i}^2}=\sum_{i=1}^{n}{a_{i}b_{i}}^2$$. Titu's Lemma: Let (\ a_{i},b_{i}\in{\mathbb{R}}) and let (\ a_{i},b_{i}>0) for (\ i\in{1,2,\dots,n}) $$\sum_{i=1}^{n}\frac{a_{i}^2}{b_{i}}\ge\frac{{\sum_{i=1}^{n}a_{i}}^2}{\sum_{i=1}^{n}b_{i}}$$ Proof of Cauchy Schwarz's Inequality: We […]

Problem 1 Find the sum 𝑆=Σ2015𝑘=1(−1)𝑘(𝑘+1)2⋅𝑘 Problem 2 Suppose in $\triangle A B C$, $A B=\sqrt{3}$, $B C=1$, $C A=2$. Suppose there exists a point $P_{0}$ in the plane of $\triangle A B C$ such that $A P_{0}$+$B P_{0}$+$C P_{0} \leq A P+B P+C P$ for all points $P$ in the plane of $\triangle A […]

Here is a variant of clueless Sudoku that I was trying for fun. In clueless Sudoku A n*n board is given No numbers are written on the board The board is divided in some blocks (typically in some pattern). Sum of numbers in each block must be constant. Numbers 1 to n must appear in […]

What is the value of \( \dfrac{11!-10!}{9!}\)? (A) 99 (B) 100 (C) 110 (D) 121 (E) 132 For what value of \( x \) does \( 10^x \cdot 100^{2x} = 1000^5 \)? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $12.50 more than David. […]

Try the solution of problem from RMO (Regional Mathematical Olympiad) 2015 Problem 1 based on Triangle. Problem: Triangle Problem Two circles $latex \Gamma $ and $latex \Sigma $, with centers O and O', respectively, are such that O' lies on $latex \Gamma $. Let A be a point on $latex \Sigma $, and let M […]

Try this problem from RMO 2015 from Chennai Region based on Integer Solution of Polynomial. Problem: Integer Solution of Polynomial Solve the equation $latex y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20 &s=2 $ for positive integers x, y. Discussion: $latex y^3 + 3y^2 + 3y = x^3 + 5x^2 […]

This is a problem from RMO 2015 Mumbai Region based on inequality. Problem: RMO 2015 Mumbai Region Let x, y, z be real numbers such that $ x^2 + y^2 + z^2 - 2xyz = 1 $ and $ s=2$ . Prove that $ (1+x)(1+y)(1+z) \le 4 + 4xyz $ and $ s=2$ Discussion Note […]

This is a problem from RMO 2015 from Mumbai Region based on Cyclic Quadrilaterals and Incenters. Problem: RMO 2015 Mumbai Region Let ABC be a right angled triangle with $ \angle B = 90^0 $ and $ s=2 $ and let BD be the altitude from B on to AC. Draw $ DE \perp AB […]

This is a problem from Regional Mathematics Olympiad, RMO 2015 Mumbai Region based on Polynomial with positive integers. Try to solve it. Site title Title Primary category Separator Problem: Let P(x) be a polynomial whose coefficients are positive integers. If P(n) divides P(P(n) -2015) for every natural number n, prove that P(-2015) = 0. Discussion:  Let […]