What is the sum of this series up to its 100th term? $$\frac25+\frac15+\frac2{15}+\frac1{15}+\frac1{15}+\frac1{30}+\frac1{25}+\frac1{50}+\frac2{75}+\cdots$$
Pattern convention: continue the paired reciprocal-product pattern.
The passcode to Nathalie’s computer is described in a hint: In a collection of 9001 distinct objects, there are 𝑁 ways to arrange their order. The number of zeros at the end of 𝑁 is the passcode. What is the passcode?
Consider the expression $\frac{2^{k_1}-2^{k_2}}{2^{k_3}-2^{k_4}}$ where $k_1,k_2,k_3,k_4$ are all non-negative integers. What is the smallest value of $n$ where $n$ represents the positive integers that cannot be equal to the expression?
Families living in New Clark City love biking. There are a total of 2,022 families in the city each with two or four bicycles. Given that there are as many families owning either four bicycles as many as families owning two bicycles, how many bicycles are there in New Clark City?
Vladimir’s wallet contains 20, 50, and 100-peso bills. If the total value of the bills is Php1,650 and there are 30 bills, how many combinations of bills are possible?
Interpretation: at least one bill of each listed denomination is present.
Editorial note: the printed target CVCC differs from the result label CVVC in the original figure. Answer the printed target. Distinct letters denote distinct decimal digits; leading digits are nonzero.
The radius of circle A is 1/3 the radius of circle B. Circle A rolls around circle B one trip back to its starting point. How many times will circle A rotate in total?
Convention: external rolling without slipping; circle B is fixed.
Let $e$ be Euler’s constant, an irrational number equal to $2.71828\ldots$. Let $e^2$ and $e$ be the roots of $x^2-bx+c=0$. Find the real number $b+c$.
Input format: use pi, sqrt(...), ^ for powers, * for multiplication and / for division as needed.
Let $R$ and $F$ ($R,F\ne0$) be the roots of the equation $x^2-3x-m=0$. If $RF$ and $R+F$ are the roots of the equation $(c+1)x^2-sx+m=0$ and $c+1+s+m=0$, then find the value of $c+s-m$.
Find $\sqrt{S+\frac1{10^9}}$ if $$S=\frac{10}{100}+\frac{200}{100^2}+\frac{3000}{100^3}+\frac{40000}{100^4}+\cdots+\frac{9000000000}{100^9}+\frac{100000000000}{100^{10}}.$$
Input format: use pi, sqrt(...), ^ for powers, * for multiplication and / for division as needed.
Hayley fills a spherical fishbowl with water up to the level indicated in the figure below. The radius of the entire sphere is 15 decimeters. If the height of the unfilled portion is h=3 decimeters while radius r=9 decimeter, what is the volume of the water in the fishbowl? (Leave π as it is or express your answer using π)
Input format: use pi, sqrt(...), ^ for powers, * for multiplication and / for division as needed.
In the given problem in #19, what is the surface area of the portion of the fishbowl that is filled with water? (Leave π as it is or express your answer using π.)
Context: the sphere has radius 15 dm, and the empty top cap has height 3 dm and rim radius 9 dm. Interpret surface area as the wetted curved wall. Input format: use pi, sqrt(...), ^ for powers, * for multiplication and / for division as needed.
