Search problems

5 problems match “integer constraints”. Clear

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

1. Largest sum for an involutory matrix ●●○○○ 20 ptsLet $a,$ $b,$ $c$ be integers such that $\mathbf{A} = \frac{1}{5} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix}$ and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$
2. Largest number of jogs ●●○○○ 20 ptsBill can buy jags, jigs and jogs for $1, $2 and $7 each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $50?
3. Largest integer in a five-number list ●●○○○ 20 ptsA list of five positive integers has all of the following properties: - The only integer in the list that occurs more than once is $8$. - Its median is $9$. - Its average (mean) is $10$. What is the largest possible integer that could appear in the list?
4. Smallest survey size with given percentages ●●○○○ 20 ptsA total of $n$ people were asked a question in a survey. Exactly $76\%$ of the $n$ people responded "yes" and exactly $24\%$ of the $n$ people responded "no". What is the smallest possible value of $n$?
5. Minimum respondents on the twelfth day ●●●○○ 50 ptsDuring the first eleven days, 700 people responded to a survey question. Each person chose exactly one of three options, and the ratio of the numbers of responses was $4:7:14$. On the twelfth day, several more people responded, so that the number of responses for each option did not decrease and the ratio became $6:9:16$. What is the minimum number of people who must have responded on the twelfth day?