2019 πtagoras Oral Rounds 1

Scoring Scheme
# of Teams Correct # of Points
1 to 3 4
4 to 7 3
8 to 10 2

Mental Round
M1 (20s) Let p be a cubic polynomial with the following properties:
I The leading coefficient of p is 1.
II All the roots of p are positive real numbers.
III p(0) = −1000.
Find the largest possible value of p(−1). [−1331]
3
M2 (60s) Given that 2(5x+1 ) = 1 + , find all possible values of x. Express your answer/s in the form
5x
a + log5 b where a and b are integers. [−1 + log5 3]
√
M3 (20s) Suppose f ∶ R → R such that f (a + b) = f (ab) for all a, b ∈ R, and f (673) = − 3 4. Find √
f (2019). [− 3 4]
50 √
M4 (60s) Given a real number x, define ⌊x⌋ as the largest integer less than or equal to x. Calculate ∑ ⌊ k⌋.
k=1
[217]
M5 (40s) A nondegenerate triangle with integral sides has a perimeter and area A. What are all the possible
√
values of A? [2 2]
2019
M6 (20s) Determine the ones digit of −1 + ∑ k 5 . [9]
k=1

M7 (40s) How many three-digit natural numbers ust represent a valid date in 2019, where u or us corre-
sponds to a month and st or t, respectively, corresponds to a day in that a month? For example, 105 is
a valid representation for either January 5 or October 5. [273]
√ 2019
M8 (20s) Let i = −1 and Re(a + bi) = a. Compute ∑ Re(i k ). [−1]
k=1

M9 (20s) We say that a 5-card hand is monarchial if the hand contains only queens and kings. How many
monarchial hands are there in a standard 52-card deck? [56]
M10 (20s) What is the largest integer n such that 2n ≤ n3 ? [9]
9
M11 (40s) Find the greatest possible value of k if x(3 − 2x) = 4k has at least one real solution. [ ]
32
n! n! n! n! n!
M12 (20s) Mark continues to write the terms of the sequence , , , , . . . so long as is an integer.
1 2 3 4 k
How many terms will Mark write if n = 90? [96]
 2019 πtagoras Oral Rounds 2

⎧
⎪
⎪1, if x < 0
M13 (60s) Define f (x) = 1 − (−1)x for every real number x and g(x) = ⎨ . Denote the compo-
⎪
⎪ −1, if x ≥ 0
⎩
2019
sition of f with itself n times as f n (x) = ( f ○ f ○⋯○ f )(x). Find the sum ∑ ( f ○ g)k ((−1)k ). [4038]
k=1

M14 (40s) Given√ a real number x, denote ⌊x⌋ as the largest integer less than or equal to x. If for every positive
integer n, ⌊ n2 + n⌋ = an2 + bn + c where a, b and c are real numbers, find the arithmetic mean of a,
1
b and c. [ ]
3
M15 (10s) He is a British-Lebanese algebraic topologist, geometer, and Fields Medalist who claimed to have
proven the Riemann hypothesis in September 2018 at the 2018 Heidelberg Laureate Forum. Unfortu-
nately, his proof has been rejected by the mathematical community. Who is he? (Surname will do.)
[Sir Michael Atiyah]
Written Round
√
W1 (30s) The sum of the arithmetic mean and the geometric mean of two positive numbers is 2. What
√
is the sum of the positive square roots of the two numbers? [ 4 8]
W2 (30s) Given f (x) = x 2 + x(2 − k) + k 2 , find the range of values of k for which f (x) > 0 for all real values
2
x. [k > or k < −2]
3
W3 (180s) Find the number of sequences of length 2019 containing only letters from the English alphabet
2044 2044
(with 26 letters) that have the letters in alphabetical order. [( ) or ( )]
25 2019
W4 (120s) Given that three roots of f (x) = x 4 + x 3 + ax 2 + bx + c are 1, −3 and 5, what is the value of
a + b + c? [−2]
W5 (180s) What is the side length of the smallest square that can enclose 3 non-overlapping unit
√ circles?
√
2 6
[2 + + ]
2 2
W6 (40s) Express the finite product (x + 1)(x 2 + 1)(x 4 + 1)⋯(x 1024 + 1) for all real values x as a finite sum.
2047
Express your answer in sigma (summation) notation.* [1 + ∑ x k ]
k=1

W7 (90s) Let a1 , a2 , a3 , . . . be a sequence of numbers such that a n = a n−1 + a n−2 for every integer n ≥ 3. If
a7 = 69, calculate the sum of the first 10 terms of the sequence. [759]
W8 (90s) Let S = {1, 2, 3, 4, . . . , 50} be the set containing the least 50 positive integers. If a set X is chosen
randomly from the power set of S, what is the probability that X contains an integer relatively prime
1
to 50? [1 − 20 ]
2
(n + 1)2
W9 (90s) For how many natural numbers n is an integer? [4]
n + 23
W10 (30s) If all square numbers are removed from the sequence 1, 2, 3, 4, . . . , 2019, what is the 1611th term?
[1651]
W11 (180s) A polynomial is said to be cute if it satisfies the following:
* Voided and replaced with a clincher problem (problem 26).
 2019 πtagoras Oral Rounds 3

I It is quadratic with integer coefficients.

II Its leading coefficient is 1.
III One of its roots is 11.
IV Its roots differ by no more than 11.
What is the sum of the coefficients of all cute polynomials? [2300]
W12 (120s) A hexagon is inscribed in a circle of radius 1. Each of the alternate sides of the hexagon has
length 1. Let A, B, and C be the midpoints of the three other sides. If line segment AB has length
√ x,
3 2
what is the area of triangle ABC in terms of x. [ x ]
4
W13 (40s) In 2018, Xizelle has exactly one pair of shoes. As her New Year’s resolution for 2019, she will
double the number of shoes she has every year by splurging in Shopee. Assuming that her shoes come
in matching pairs and no two pairs are identical, determine the earliest year when Xizelle has the ability
to wear at least 2019 mismatched pairs of shoes. Note that left and right shoes are distinguishable, and
she must always wear one of each. [2024]
W14 (45s) Shown in the figure are two concentric circles and a line tangent to the inner circle. The tangent
line has a length ℓ, and meets the outer circle at A and B. What is the area of the annulus (shaded
1
region) in terms of ℓ? [ πℓ2 ]
4
B

√
W15 (90s) In triangle U ST, U T = 9 and P lies on U S such that U P = SP = TP = 5. What is ST? [ 19]
W16 (180s) Given the recurrence relation y n+2 − 2y n+1 − 2y n = 0 where √y1 = 1, y2 = 3, and n is a positive
3 √ √
integer, find y n in terms of n. [y n = ((1 + 3)n+1 − (1 − 3)n+1 )]
12
W17 (180s) Simplify 1 + 2 ⋅ 2 + 3 ⋅ 22 + 4 ⋅ 23 + 5 ⋅ 24 + ⋯ + 2019 ⋅ 22018 . [2018 ⋅ 22019 + 1]
√
W18 (180s) If log4 (x + 2y) + log4 (x − 2y) = 1, find the minimum value of ∣x∣ − ∣y∣. [ 3]
W19 (90s) Consider the sequence given by a n = a n−1 + 3a n−2 + a n−3 , where a0 = a1 = a2 = 1. What is the
remainder when a2019 is divided by 7? [5]
W20 (60s) Let θ be an acute angle such that the equation x 2 + 4x cos θ + cot θ = 0 involving the variable x
π 5π
has multiple roots. Find the possible values of the measure of θ in radians. [ <x< ]
12 12
W21 (120s) Find the ordered triples of real numbers (u, s, t) such that u + s + t > 2, u 2 + s 2 = 4 − 2us,
1 3 5 3 1 9
u 2 + t 2 = 9 − 2ut, and s2 + t 2 = 16 − 2ut. [( , , ) and (− , , )]
2 2 2 2 2 2
 2019 πtagoras Oral Rounds 4

W22 (180s) Greg flips a fair coin 2019 times while Ian flips a fair coin 2020 times. What is the probability
1
that Greg gets at least as many heads as Ian? [ ]
2
W23 (120s) PTGR is a regular tetrahedron with side length 2020. What is the area of the cross section of
PTGR cut by the plane that passes through the midpoints PT, PG, and GR? [1 020 100]
W24 (60s) Alvin wants to run 4 laps around their village at an average speed of 10 km/h. If he completes the
first 3 laps at an average speed of only 9 km/h, what should his average speed be on his fourth lap to
achieve his goal? [15 km/h]
W25 (180s) Let n be a base-10 number. The value of n when interpreted as a base-20 number is twice the
value of n when interpreted as a base-13 number. Calculate the sum of all possible values of n.
[198]
√ √ 546
W26 (180s) Find all real numbers x such that 20x + 3 20x + 13 = 13. [ ]
3