A dog barks every $b$ minutes and a cat meows every $m$ minutes. Both $b,m$ are primes with $1\le b\le10$ and $10\le m\le20$. What is the greatest number of simultaneous calls in a 2-hour period? To make the time convention explicit, both call at time 0; count only calls with $0<t\le120$ minutes.
Emily and Hannah initially had the same number of tickets. Emily gave Hannah 5 tickets, then Hannah bought 2 more. At the end Hannah had four times as many tickets as Emily. How many tickets did Emily have initially?
Andy and Brett start at opposite points on a circular track of circumference 900 feet and jog in the same direction shown. Andy runs at 4.3 feet per second and Brett at 5.5 feet per second. How many seconds pass before they first meet?
A wire is folded into a square of area $324\text{ cm}^2$. It is straightened, cut into equal lengths and fastened to make the frame of a cube. No wire is wasted. Find the volume enclosed by the cube, in cubic centimetres.
Mel sells jumbo cheesy pretzel nuggets in packs of 5 (“diet”), 6 (“share”), and 11 (“party”). What is the greatest total number of nuggets that cannot be bought exactly? For example, 15 can be bought but 7 cannot. Any nonnegative whole number of each pack may be bought.
Joe and Tracy have 26 pieces of candy altogether. Joe gives Tracy $n$ pieces so that she has twice as many pieces as Joe initially had. How many different values of $n$ are possible? Candy counts are whole numbers; $n$ is positive and Joe may give away all his candy.
The absolute value $|N|$ is the distance from $N$ to 0 on the number line, regardless of sign; for example, $|+5|=5$ and $|-3|=3$. Find the sum of all integer values of $N$ satisfying $$15>\frac{30}{|N-3|}>6.$$ The expression is defined only when $N$ is different from 3.
Betsy collects sports posters. Exactly one of these statements is false:
- Betsy has at least 122 sports posters.
- Betsy has at least 131 sports posters.
- Betsy has at least 135 sports posters.
- Betsy has at least 139 sports posters.
What is the greatest possible number of posters in her collection?
Let $N$ be a whole number (a nonnegative integer). Exactly three of these four statements are true:
1. $N$ is prime.
2. $N+3$ is prime.
3. $1<\sqrt N<9$.
4. $N+5$ has an odd number of positive factors.
A number has all its digits equal to 1 in base 3, and its last digit in base 2 is 0. What is the largest four-digit number in base 10 with this property?
