Let \(XZ\) be a diameter of circle \(\omega\). Let Y be a point on \(XZ\) such that \(XY=7\) and \(YZ=1\). Let W be a point on \(\omega\) such that \(WY\) is perpendicular to \(XZ\). What is the square of the length of the line segment \(WY\) ?
(a) 7
(b) 8
(c) 10
(d) 25
Answer A
How many five-digit numbers containing each of the digits 1,2,3,4,5 exactly once are divisible by 24 ?
(a) 8
(b) 10
(c) 12
(d) 20
Answer B
A lattice point is a point (x, y) where x and y are both integers. Find the number of lattice points that lie on the closed line segment whose endpoints are (2002,2022) and (2022,2202).
(a) 20
(b) 21
(c) 22
(d) 23
Answer B
Let \(\omega \neq-1\) be a complex root of \(x^3+1=0\). What is the value of \(1+2 \omega+3 \omega^2+4 \omega^3+5 \omega^4\) ?
(a) 3
(b) -4
(c) 5
(d) -6
Answer D
How many ending zeroes does the decimal expansion of \(2022!\) have?
(a) 404
(b) 484
(c) 500
(d) 503
Answer D
Two tigers, Alice and Betty, run in the same direction around a circular track of circumference 400 meters. Alice runs at a speed of \(10 \mathrm{~m} / \mathrm{s}\) and Betty runs at \(15 \mathrm{~m} / \mathrm{s}\). Betty gives Alice a 40 meter headstart before they both start running. After 15 minutes, how many times will they have passed each other?
(a) 9
(b) 10
(c) 11
(d) 12
Answer D
Suppose a, b, c are the roots of the polynomial \(x^3+2 x^2+2\). Let f be the unique monic polynomial whose roots are \(a^2, b^2, c^2\). Find \(f(1)\). (Note: A monic polynomial is a polynomial whose leading coefficient is 1 .)
(a) -17
(b) -16
(c) -15
(d) -14
Answer C
Let I be the center of the incircle of triangle ABC. Suppose that this incircle has radius 3 , and that A I=5. If the area of the triangle is 2022 , what is the length of B C ?
(a) 670
(b) 672
(c) 1340
(d) 1344
Answer A
A square is divided into eight triangles as shown below. How many ways are there to shade exactly three of them so that no two shaded triangles share a common edge?
(a) 12
(b) 16
(c) 24
(d) 30
Answer B
The numbers 2, b, c, d, 72 are listed in increasing order so that 2, b, c form an arithmetic sequence, b, c, d form a geometric sequence, and c, d, 72 form a harmonic sequence (that is, a sequence whose reciprocals of its terms form an arithmetic sequence). What is the value of b+c ?
(a) 7
(b) 13
(c) 19
(d) 25
Answer C
How many positive integers \(n<2022\) are there for which the sum of the odd positive divisors of n is 24 ?
(a) 7
(b) 8
(c) 14
(d) 15
Answer D
Call a whole number ordinary if the product of its digits is less than or equal to the sum of its digits. How many numbers from the set \({1,2, \ldots, 999}\) are ordinary?
(a) 151
(b) 162
(c) 230
(d) 241
Answer D
What is the area of the shaded region of the square below?
(a) 7
(b) 11
(c) 15
(d) 19
Answer D
Bryce plays a game in which he flips a fair coin repeatedly. In each flip, he obtains two tokens if the coin lands on heads, and loses one token if the coin lands on tails. At the start, Bryce has nine tokens. If after nine flips, he also ends up with nine tokens, what is the probability that Bryce always had at least nine tokens?
(a) \(1 / 7\)
(b) \(1 / 6\)
(c) \(5 / 28\)
(d) \(17 / 84\)
Answer A
How many ways are there to arrange the first ten positive integers such that the multiples of 2 appear in increasing order, and the multiples of 3 appear in decreasing order?
(a) 720
(b) 2160
(c) 5040
(d) 6480
Answer D
What is the largest multiple of 7 less than 10,000 which can be expressed as the sum of squares of three consecutive numbers?
Suppose that the polynomial \(P(x)=x^3+4 x^2+b x+c\) has a single root r and a double root s for some distinct real numbers r and s. Given that \(P(-2 s)=324\), what is the sum of all possible values of \(|c|\) ?
Let m and n be relatively prime positive integers. If \(m^3 n^5\) has 209 positive divisors, then how many positive divisors does \(m^5 n^3\) have?
Let x be a positive real number. What is the maximum value of \(\frac{2022 x^2 \log (x+2022)}{(\log (x+2022))^3+2 x^3}\)?
Let a, b, c be real numbers such that
\(3 a b+2=6 b, \quad 3 b c+2=5 c, \quad 3 c a+2=4 a\).
Suppose the only possible values for the product a b c are \(r / s\) and \(t / u\), where \(r / s\) and \(t / u\) are both fractions in lowest terms. Find \(r+s+t+u\).
You roll a fair 12 -sided die repeatedly. The probability that all the primes show up at least once before seeing any of the other numbers can be expressed as a fraction \(p / q\) in lowest terms. What is \(p+q\) ?
Let PMO be a triangle with PM=2 and \(\angle PMO=120^{\circ})\). Let B be a point on PO such that PM is perpendicular to MB, and suppose that PM=BO. The product of the lengths of the sides of the triangle can be expressed in the form \(a+b \sqrt[3]{c}\), where a, b, c are positive integers, and c is minimized. Find a+b+c.
Let ABC be a triangle such that the altitude from A, the median from B, and the internal angle bisector from C meet at a single point. If BC=10 and CA=15, find \(AB^2\).
Find the sum of all positive integers \(n, 1 \leq n \leq 5000\), for which
\(n^2+2475 n+2454+(-1)^n\)
is divisible by 2477 . (Note that 2477 is a prime number.)
For a real number x, let \(\lfloor x\rfloor\) denote the greatest integer not exceeding x. Consider the function
\(f(x, y)=\sqrt{M(M+1)}(|x-m|+|y-m|),\)
where \(M=\max (\lfloor x\rfloor,\lfloor y\rfloor)\) and \(m=\min (\lfloor x\rfloor,\lfloor y\rfloor)\). The set of all real numbers \((x, y)\) such that \(2 \leq x, y \leq 2022\) and \(f(x, y) \leq 2\) can be expressed as a finite union of disjoint regions in the plane. The sum of the areas of these regions can be expressed as a fraction \(a / b\) in lowest terms. What is the value of a+b ?