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14 problems match “inequalities”. Clear

Average of integers between two fractions ●○○○○ 10 ptsWhat is the average of all the integer values of $M$ such that $\frac{M}{56}$ is strictly between $\frac{3}{7}$ and $\frac{1}{4}$ ?
Blue points on a line ●●○○○ 20 ptsOn a line, one hundred points are marked, each painted green, blue, or red. Between any two red points there is a blue point, and between any two blue points there is a green point. There are no fewer red points than blue points, and no fewer blue points than green points. How many points are painted blue?
Candle height at half burn time ●●○○○ 20 ptsA large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$ -th centimeter. (The candle burns down each individual centimeter at a fixed rate.) Suppose it takes $T$ seconds for the candle to burn down completely. Compute the candle's height in centimeters $\tfrac{T}{2}$ seconds after it is lit.
Trash can unit prices and minimum type A quantity ●●○○○ 20 ptsA community buys two models of trash cans, A and B. Buying 3 type-A and 4 type-B cans costs a total of $580, and buying 6 type-A and 5 type-B cans costs a total of $860. (1) Find the unit price of a type-A can and of a type-B can. (2) A total of 200 cans of types A and B are needed, and the total cost must not exceed $15000. What is the least number of type-A cans that should be purchased?
Possible values of f(2018) ●●●○○ 50 ptsLet $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be an even function with $f(1)=1$ and $f(2017) \neq 1$ . For all integers $x, y$ , $2f(x+y)-f(x)-f(y) \leqslant |f(x)-f(y)|$ . Find the value of $f(2018)$ .
Square root plus arcsine value ●●●○○ 50 ptsGiven a real number $x$ , find the value of $\sqrt{x(x+1)}+\arcsin \sqrt{x^{2}+x+1}$ , in radians. Round to two decimal places.
Smallest bound on derivative at zero ●●●○○ 50 ptsFind the smallest real number $A$ such that for every quadratic polynomial $f(x)$ satisfying $|f(x)| \leqslant 1$ for all $0 \leqslant x \leqslant 1$ , the inequality $f'(0) \leqslant A$ holds.
Smallest bound on function differences ●●●○○ 50 ptsThe function $f(x)$ is defined on $[0,1]$ and satisfies $f(0)=f(1)$ , and for any $x, y \in [0,1]$ with $x \neq y$ , we have $|f(x)-f(y)|<|x-y|$ . Find the smallest real number $m$ such that for every such function $f$ and any $x, y \in [0,1]$ , we have $|f(x)-f(y)|<m$ . Give your answer as a decimal.
Largest edge of a box ●●●○○ 50 ptsAn $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are positive integers with $m\le n\le p$ . Find the largest possible value of $p$ .
Product preserved after altering two numbers ●●●○○ 50 ptsLucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, and finds that the product of the two original numbers equals the product of the two altered numbers. Compute the minimum possible sum of the original two numbers.