Suppose that the polynomial \(f\left(x\right)=2x^5-9x^3+2x^2+9x-3\) has 5 solutions \(x_1;x_2;x_3;x_4;x_5\). The other polynomial \(K\left(x\right)=x^2-4\). Find the value of \(P=K\left(X_1\right)\times K\left(X_2\right)\times K\left(X_3\right)\times K\left(X_4\right)\times K\left(X_5\right)\)
Edna enters a room with 1000 bottles lined up in a row left to right. One bottle contains a tasteless magic potion. All bottles to the left of the magic potion contain tasteless water. All bottles to the right of the magic potion contain a bitter poison. Edna can drink from no more than two bottles containing poison before becoming sick and being unable to drink anything else. She can take an unlimited number of drinks from any other bottle. What is the minimum number of bottles from which Edna may need to drink to ensure she can identify the bottle containing the magic potion no matter where it is in the lineup?
What is the last 3-digit A value of the board?
Penner has a deck of 40 cards composed of four suits (red, blue, green, and yellow) and cards numbered 1 through 10 in each suit. Tell secretly chooses a card. Penner then chooses the following 4 cards from the deck: Red-2, Blue-3, Green-5 and Yellow-7. For each card Penner chooses, Tell says “yes” if his secret card is of the same color or shares a common factor greater than 1 with Penner’s card. Otherwise Tell says “no.” Tell says “no,” “yes,” “no,” and “yes,” respectively, in response to Penner’s cards. With this information, what is the best possible probability Penner has of guessing Tell’s secret card? Express your answer as a common fraction.
A set S contains some, but not all, of the positive integers from 3 to 7. Some statements describing S are given below. The statement numbered n is true if the number n is in S and false if n is not in S. What is the product of the numbers that are in S?
3. The sum of the numbers in S is odd.
4. The sum of the numbers in S is less than 15.
5. S contains exactly one composite number.
6. S contains exactly one prime number.
7. The product of the numbers in S is odd.