In a convex polygon, the number of diagonals is 23 times the number of its sides. How many sides does it have?
(a) 46
(b) 49
(c) 66
(d) 69
Answer: B
What is the smallest real number a for which the function \(f(x)=4 x^2-12 x-5+2a\) will always be nonnegative for all real numbers x ?
(a) 0
(b) \(\frac{3}{2}\)
(c) \(\frac{5}{2}\)
(d) 7
Answer: D
In how many ways can the letters of the word PANACEA be arranged so that the three As are not all together?
(a) 540
(b) 576
(c) 600
(d) 720
Answer: D
How many ordered pairs of positive integers \((x, y)\) satisfy \(20 x+21 y=2021\) ?
(a) 4
(b) 5
(c) 6
(d) infinitely many
Answer: B
Find the sum of all k for which \(x^5+k x^4-6 x^3-15 x^2-8 k^3 x-12 k+21\) leaves a remainder of 23 when divided by \(x+k\).
(a) -1
(b) \(9-\frac{3}{4}\)
(c) \(\frac{5}{8}\)
(d) \(\frac{3}{4}\)
Answer: B
In rolling three fair twelve-sided dice simultaneously, what is the probability that the resulting numbers can be arranged to form a geometric sequence?
(a) \(\frac{1}{72}\)
(b) \(\frac{5}{288}\)
(c) \(\frac{1}{48}\)
(d) \(\frac{7}{288}\)
Answer: D
How many positive integers n are there such that \(\frac{n}{120-2 n}\) is a positive integer?
(a) 2
(b) 3
(c) 4
(d) 5
Answer: B
Three real numbers \(a_1, a_2, a_3\) form an arithmetic sequence. After \(a_1\) is increased by 1 , the three numbers now form a geometric sequence. If \(a_1\) is a positive integer, what is the smallest positive value of the common difference?
(a) 1
(b) \(\sqrt{2}+1\)
(c) 3
(d) \(\sqrt{5}+2\)
Answer: B
Point G lies on side A B of square A B C D and square A E F G is drawn outwards A B C D, as shown in the figure below. Suppose that the area of triangle E G C is \(1 / 16\) of the area of pentagon D E F B C. What is the ratio of the areas of A E F G and A B C D ?
(a) \(4: 25\)
(b) \(9: 49\)
(c) \(16: 81\)
(d) \(25: 121\)
Answer: A
In how many ways can 2021 be written as a sum of two or more consecutive integers?
(a) 3
(b) 5
(c) 7
(d) 9
Answer: C
In quadrilateral \(A B C D, \angle C B A=90^{\circ}, \angle B A D=45^{\circ}\), and \(\angle A D C=105^{\circ}\). Suppose that \(B C=1+\sqrt{2}\) and \(A D=2+\sqrt{6}\). What is the length of A B ?
(a) \(2 \sqrt{3}\)
(b) \(2+\sqrt{3}\)
(c) \(3+\sqrt{2}\)
(d) \(3+\sqrt{3}\)
Answer: C
Alice tosses two biased coins, each of which has a probability p of obtaining a head, simultaneously and repeatedly until she gets two heads. Suppose that this happens on the r th toss for some integer \(r \geq 1\). Given that there is \(36 \%\) chance that r is even, what is the value of p ?
(a) \(\frac{\sqrt{7}}{4}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{\sqrt{2}}{2}\)
(d) \(\frac{3}{4}\)
Answer: A
For a real number \(t,\lfloor t\rfloor\) is the greatest integer less than or equal to t and \({t}=t-\lfloor t\rfloor\) is the fractional part of t. How many real numbers x between 1 and 23 satisfy \(\lfloor x\rfloor{x}=2 \sqrt{x}\) ?
(a) 18
(b) 19
(c) 20
(d) 21
Answer: A
Find the remainder when \(\sum_{n=2}^{2021} n^n\) is divided by 5 .
(a) 1
(b) 2
(c) 3
(d) 4
Answer: D
In the figure below, B C is the diameter of a semicircle centered at O, which intersects A B and A C at D and E respectively. Suppose that \(A D=9, D B=4\), and \(\angle A C D=\angle D O B\). Find the length of A E.
(a) \(\frac{117}{16}\)
(b) \(\frac{39}{5}\)
(c) \(2 \sqrt{13}\)
(d) \(3 \sqrt{13}\)
Answer: B
Consider all real numbers c such that \(|x-8|+\left|4-x^2\right|=c\) has exactly three real solutions. The sum of all such c can be expressed as a fraction \(a / b\) in lowest terms. What is \(a+b\) ?
Answer: 93
Find the smallest positive integer n for which there are exactly 2323 positive integers less than or equal to n that are divisible by 2 or 23 , but not both.
Answer: 4644
Let \(P(x)\) be a polynomial with integer coefficients such that \(P(-4)=5\) and \(P(5)=-4\). What is the maximum possible remainder when \(P(0)\) is divided by 60 ?
Answer: 41
Let \(\triangle ABC\) be an equilateral triangle with side length 16. Points D, E, F are on C A, A B, and B C, respectively, such that \(DE \perp AE, DF \perp CF\), and \(BD=14\). The perimeter of \(\triangle BEF\) can be written in the form \(a+b \sqrt{2}+c \sqrt{3}+d \sqrt{6}\), where a, b, c, and d are integers. Find \(a+b+c+d\).
Answer: 31
How many subsets of the set \({1,2,3, \ldots, 9}\) do not contain consecutive odd integers?
Answer: 208
For a positive integer n, define \(s(n)\) as the smallest positive integer t such that n is a factor of t \( !.\) Compute the number of positive integers n for which \(s(n)=13\).
Answer: 792
Alice and Bob are playing a game with dice. They each roll a die six times, and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6,5, and 6 , while Bob's first three rolls are 2,1 , and 3 . The probability that Bob wins can be written as a fraction \(a / b\) in lowest terms. What is \(a+b\) ?
Answer: 3895
Let \(\triangle ABC\) be an isosceles triangle with a right angle at A, and suppose that the diameter of its circumcircle \(\Omega\) is 40 . Let D and E be points on the arc BC not containing A such that D lies between B and E, and AD and A E trisect \(\angle BAC\). Let \(I_1\) and \(I_2\) be the incenters of \(\triangle ABE\) and \(\triangle ACD\) respectively. The length of \(I_1 I_2\) can be expressed in the form \(a+b \sqrt{2}+c \sqrt{3}+d \sqrt{6}\), where \(a, b, c\), and d are integers. Find \(a+b \frac{b}{3} c+d\).
Answer: 20
Find the number of functions f from the set \(S={0,1,2, \ldots, 2020}\) to itself such that, for all \(a, b, c \in S\), all three of the following conditions are satisfied:
(i) If f(a)=a, then a=0;
(ii) If f(a)=f(b), then a=b; and
(iii) If \(c \equiv a+b(\bmod 2021)\), then f(c) \(\equiv f(a)+f(b)(\bmod 2021)\).
Answer: 1845
A sequence \(\left{a_n\right}\) of real numbers is defined by \(a_1=1\) and for all integers \(n \geq 1\),
\(a_{n+1}=\frac{a_n \sqrt{n^2+n}}{\sqrt{n^2+n+2 a_n^2}}\).
Compute the sum of all positive integers \(n<1000\) for which \(a_n\) is a rational number.
Answer: 131
