MathDB

3

Part of 2023 Tuymaada Olympiad

Problems(2)

Two angles summing to 180 degrees

Source: Tuymaada 2023 Junior P3

7/12/2023
Point LL inside triangle ABCABC is such that CL=ABCL = AB and ∠BAC+∠BLC=180∘ \angle BAC + \angle BLC = 180^{\circ}. Point KK on the side ACAC is such that KL∥BCKL \parallel BC. Prove that AB=BKAB = BK
geometrycyclic quadrilateralreflectionTuymaada
Inequality with cubic roots

Source: Tuymaada 2023 Senior P3

7/7/2023
Prove that for every positive integer n≥2n \geq 2, ∑1≤i≤nin+13n≤∑1≤i≤n−1in3n−1.\frac{\sum_{1\leq i \leq n} \sqrt[3]{\frac{i}{n+1}}}{n} \leq \frac{\sum_{1\leq i \leq n-1} \sqrt[3]{\frac{i}{n}}}{n-1}.
algebra