Let x be a positive integer whose prime factors are 5 and 11 only. If x has exactly 6 different positive integer factors, what is the smallest value of x?
Interpretation: Both 5 and 11 occur as prime factors.
Let $A,M,N,P,S,W$ be real numbers such that
$$M=S^1W^2A^3P^4,\qquad N=P^1A^2W^3S^4.$$
Suppose $W\times A\times S\times P=1331^{1/15}$. What is $M\times N$?
Interpretation: The source question has a faint/missing first factor before ×A×S×P. We restore W, as explicitly used in the source solution.
A six-digit positive integer contains the digit 8 and is divisible by 9. All its digits are different. What is the sum of the digits of the least possible such integer?
Gryffindor Secondary School has 170 students. A survey of their hobbies shows:
- 79 like alchemy, 75 like origami, and 34 like stage illusion;
- 32 like both alchemy and origami;
- 24 like both alchemy and stage illusion;
- 13 like all three;
- 45 have hobbies not listed above.
How many like both stage illusion and origami but not alchemy?
Interpretation: The 45 students have none of the three listed hobbies; counts of “both” include those liking all three.
Twelve people receive twelve different positive integers. In the first round, person 1 says “My number is bigger than 1”, person 2 says “My number is bigger than 2”, and so on, up to person 12 saying “My number is bigger than 12”.
In the second round their responses, in random order, are “My number is less than 1”, “My number is less than 2”, …, “My number is less than 12”. Each person is either always truthful or always lying. Excluding the host, what is the largest possible number of truth-tellers?
Two calendars are the same if all dates fall on the same weekdays. The calendar for 2019 is the same as that for which listed year? Years 2020, 2024, 2028 and 2032 are leap years.
In mathematics, the product of the first n positive integers is written $n!=n(n-1)\cdots1$. For example, $2!=2\times1$ and $7!=7\times6\times5\times4\times3\times2\times1$. What is the remainder when $2018!+2018$ is divided by 2019?
Answer format: an integer without units. Leading zeros are optional; the source uses four digit boxes.
In the expression below, b is a digit, $\overline{bbbbb}$ is a five-digit number, and $\overline{bb}$ is a two-digit number. Find the numerical value in simplest form.
$$\frac{(b+b)(\overline{bbbbb}-b)-b\overline{bb}}{b\overline{bb}}$$
Answer format: an integer without units. Leading zeros are optional; the source uses four digit boxes.
Find the value of
$$\frac{1-(1/2019)^2}{1+(1/2019)^2}+\frac{1-(1/2018)^2}{1+(1/2018)^2}+\cdots+\frac{1-(1/2)^2}{1+(1/2)^2}$$
$$+\frac{1-1^2}{1+1^2}+\frac{1-2^2}{1+2^2}+\cdots+\frac{1-2019^2}{1+2019^2}.$$
Answer format: an integer without units. Leading zeros are optional; the source uses four digit boxes.
