Maths Olympiad Daily Problems

If three distinct numbers a,b,c are in GP and equations ax² + 2bx + c = 0 and dx² + 2ex + f have a common root, then A) d,e,f are in AP B) d/a, e/b, f/c are in GP C) d/a, e/b, f/c are in AP D) d,ef are in GP

Let N = (A - B)/C where A = 999.............9 (16 digits) B = 555.......5 (8 digits) C=900.......04 (9 digits) Then the square of the sum of the digits of 2N is ?

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Tryy!! IOQM 2-3 marker level problem from Kel sirs native country

(1/2 - 1/3) + (1/2² - 1/(2·3) + 1/3²) + (1/2³ - 1/(2²·3) + 1/(2·3²) - 1/3³) + (1/2⁴ - 1/(2³·3) + 1/(2²·3²) - 1/(2·3³) + 1/3⁴) + ... is α/β, where α and β are co-prime, then α + 3β is equal to

A wizard has 16 magical potion bottles to arrange in a perfect circle around an altar. The bottles are categorized by type, and bottles of the same type are completely indistinguishable from one another:7 Healing Potions (Red)5 Mana Potions (Blue)4 Stamina Potions (Green)The circle must be arranged according to three strict rules:The Separation Rule: No two Stamina (Green) potions can be placed next to each other.The Cluster Rule: The Healing (Red) potions must form exactly two isolated clusters around the circle. (A cluster is a group of one or more identical potions sitting directly side-by-side with no other colors breaking them up).The Size Constraint: Each of the two Red clusters must contain at least 2 Red potions No. Of ways to unique ways to form circle

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