Search problems

6 problems match “consecutive integers”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. Alternating sum of consecutive integers ●○○○○ 10 ptsWhat is the value of the expression $100 - 99 + 98 - 97 + 96 - 95 + \cdots + 4 - 3 + 2 - 1$?
2. Smallest of five consecutive integers ●○○○○ 10 ptsThe mean (average) of 5 consecutive integers is 9. What is the smallest of these 5 integers?
3. Median of consecutive integer sets ●●○○○ 20 ptsLet $a$, $b$, and $c$ be integers with $a < b < c$. Let $S$ be the set of all integers from $a$ to $b$, inclusive, and $Q$ the set of all integers from $b$ to $c$, inclusive. The median of $S$ is $\frac{3}{4}b$ and the median of $Q$ is $\frac{7}{8}c$. Let $R$ be the set of all integers from $a$ to $c$, inclusive. What fraction of $c$ is the median of $R$? Give your answer as a decimal.
4. Expected consecutive pairs in a subset ●●○○○ 20 ptsA 25-element subset is chosen uniformly at random from $\{1,2,\ldots,100\}$. Find the expected number of pairs $(i,i+1)$ of consecutive integers for which both $i$ and $i+1$ lie in the subset.
5. Spacy subsets of twelve integers ●●●○○ 50 ptsCall a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 12\}$, including the empty set, are spacy?
6. Possible values of the largest removed integer ●●●●○ 100 ptsYann writes down the first $n$ consecutive positive integers, $1,2,3,4,\ldots,n-1,n$. He removes four different integers $p,q,r,s$ from the list. At least three of $p,q,r,s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is $89.5625$. What is the number of possible values of $s$?