Problems Problem 1 Let \( a, b, c \) be positive real numbers such that $$ \frac{a}{1+a} + \frac{b}{1+b} + \frac{c}{1+c} = 1 $$ Prove that \( abc \leq \frac{1}{8} \)
Problems Problem 1 Let \( a, b, c \) be positive real numbers such that $$ \frac{a}{1+a} + \frac{b}{1+b} + \frac{c}{1+c} = 1 $$ Prove that \( abc \leq \frac{1}{8} \)
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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 beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Algebraic value.
Try this beautiful problem from Probability based on dice from AMC-10A, 2011. You may use sequential hints to solve the problem
Try this beautiful problem from Geometry: Area of Region in a Circle from AMC-10A, 2011, Problem -18. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Positive solution.
Try this beautiful problem from Algebra based smallest positive value from PRMO 2019. You may use sequential hints to solve the problem.
Try this beautiful problem from combinatorics based on Regular Polygon from PRMO 2019. You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1996 based on Parallelogram Problem.
Try this beautiful problem from PRMO, 2019, problem-12, based on Integer Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance based on Graph in Calculus. You may use sequential hints to solve the problem.