Algebra 2.

5 AwesomeMath Summer Program 2024

Constructing Polynomials

1 Problem Set 5
Problem 1: Let a and b be real numbers, and let r, s, and t be the roots of f (x) = x3 +ax2 +bx−1.
Also, g(x) = x3 + mx2 + nx + p has roots r2 , s2 , and t2 . If g(−1) = −5, find the maximum possible
value of b.

Problem 2: Let f (x) = 3x3 − 5x2 + 2x − 6. If the roots of f are given by α, β, and γ, find

h
 2  2  2
1 1 1
+ + .
α−2 β−2 γ−2

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Problem 3: Let
1 1 1 1
A=
+ + + ,
2 3 5 9
1 1 1 1 1 1
B=

C=

Compute the value of A + B + C.

Problem 4: Let P√
+

+
+ +

+
1
eM
+
2·3 2·5 2·9 3·5 3·9 5·9
1 1
+
2·3·5 2·3·9 2·5·9 3·5·9
+
1
.
,

(x) =√0 be the polynomial equation of least possible degree, with rational
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coefficients, having 3 7 + 3 49 as a root. Compute the product of all of the roots of P (x) = 0.

Problem 5: Complex number ω satisfies ω 5 = 2. Find the sum of all possible values of

ω 4 + ω 3 + ω 2 + ω + 1.

Problem 6: We can show that if r1 , r2 , . . . , rn are the n roots of the polynomial xn − 1, then for
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all positive integers k with n ∤ k, we have

r1k + r2k + · · · + rnk = 0.

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Assuming this is true, prove that this statement also holds for negative integers k with n ∤ k.

Problem 7: Given that P (x) is the least degree polynomial with rational coefficients such that
√ √ √
P ( 2 + 3) = 2,

find P (10).

Problem 8: Let a1 , . . . , a2018 be the roots of the polynomial

x2018 + x2017 + · · · + x2 + x − 1345 = 0.

Compute
2018
X 1
.
1 − an
n=1

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Algebra 2.5 AwesomeMath Summer Program 2024
Problem 9: The √ monic
√ polynomial f has rational
√ coefficients
√ and is irreducible over the rational
numbers. If f ( 5 + 2) = 0, compute f (f ( 5 − 2)). (A polynomial is monic if its leading
coefficient is 1. A polynomial is irreducible over the rational numbers if it cannot be expressed as
a product of two polynomials with rational coefficients of positive degree. For example, x2 − 2 is
irreducible, but x2 − 1 = (x + 1)(x − 1) is not.)

Problem 10: The set of real numbers x for which

1 1 1
+ + ≥1
x − 2009 x − 2010 x − 2011
is the union of intervals of the form a < x ≤ b. What is the sum of the lengths of these intervals?

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Problem 11: If α, β, and γ are the roots of x3 − x − 1 = 0, compute

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1+α 1+β 1+γ
+ + .
1−α 1−β 1−γ

Problem 12: Together, Kenneth and Ellen pick a real number a. Kenneth subtracts a from every
thousandth root of unity (that is, the thousand complex numbers ω for which ω 1000 = 1) then

a1 a2 a3 a4 a5 1
eM
inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts a
from each, and then sums the results. This evidently gives the same answer! How many possible
values of a are there?

Problem 13: Let a1 , a2 , a3 , a4 , a5 be real numbers satisfying the following equations:

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+ 2 + 2 + 2 + 2 = 2 for k = 1, 2, 3, 4, 5.
k2 +1 k +2 k +3 k +4 k +5 k
a1 a2 a3 a4 a5
Find the value of 37 + 38 + 39 + 40 + 41 (Express the value in a single fraction.)

Problem 14: Let a, b, c be positive real numbers such that a + b + c = 10 and ab + bc + ca = 25.
Let m = min{ab, bc, ca}. Find the largest possible value of m.
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Problem 15: Let r1 , r2 , . . ., r20 be the roots of the polynomial x20 − 7x3 + 1. If
1 1 1
+ 2 + ··· + 2
r12 + 1 r2 + 1 r20 + 1
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m
can be written in the form n where m and n are positive coprime integers, find m + n.

Problem 16: Show that set of real numbers x which satisfy the inequality
70
X k 5
≥
x−k 4
k=1

is a union of disjoint intervals, the sum of whose lengths is 1988.

√ √
Problem 17: Is there an integer n such that n − 1 + n + 1 is a rational number?

Problem 18: Let z1 , z2 , z3 , . . . , z2021 be the roots of the polynomial z 2021 + z − 1. Evaluate
z13 z23 z33 3
z2021
+ + + ··· + .
z1 + 1 z2 + 1 z 3 + 1 z2021 + 1

2
Algebra 2.5 AwesomeMath Summer Program 2024
Problem 19: Find√ a nonzero
√ monic polynomial P (x) with integer coefficients and minimal degree
3 3
such that P (1 − 2 + 4) = 0.
Problem 20: Determine w2 + x2 + y 2 + z 2 if
x2 y2 z2 w2
+ + + =1
22 − 1 22 − 32 22 − 52 22 − 72
x2 y2 z2 w2
+ + + =1
42 − 1 42 − 32 42 − 52 42 − 72
x2 y2 z2 w2
+ + + =1
62 − 1 62 − 32 62 − 52 62 − 72
x2 y2 z2 w2

h
+ + + = 1.
82 − 1 82 − 32 82 − 52 82 − 72
Problem 21: Let a, b, c, d be real numbers such that b − d ≥ 5 and all zeros x1 , x2 , x3 , and x4

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of the polynomial P (x) = x4 + ax3 + bx2 + cx + d are real. Find the smallest value the product
(x21 + 1)(x22 + 1)(x23 + 1)(x24 + 1) can take.
Problem 22: Determine the largest real number z such that

and x, y are also real. eM

x+y+z =5
xy + yz + xz = 3

Problem 23: With all angles measured in degrees, the product

m and n are integers greater than 1. Find m + n.
Q45
k=1 csc
2 (2k − 1)◦ = mn , where
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Problem 24: We state the Principle of Inclusion and Exclusion here for the benefit of those that
are not familiar: given n finite sets A1 , A2 , . . . , An , we can calculate the size of their union as
n
!  
X X
|A1 ∪ A2 ∪ · · · ∪ An | = |Ai | −  |Ai ∩ Aj | + · · · + (−1)n+1 |A1 ∩ · · · ∩ An |
i=1 1≤i<j≤n
 
n
es

X X
= (−1)k+1 |Ai1 ∩ Ai2 ∩ · · · ∩ Aik | .
k=1 1≤i1 <···<ik ≤n

To prove this, we make an indicator function σS (x) defined on sets S and objects x as
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(
1 if x ∈ S,
σS (x) =
0 if x ̸∈ S.
P
Convince yourself that σS (x) = |S|, where x is taken over all objects.
x
(a) Prove that σA (x)σB (x) = σA∩B (x).
(b) Consider a collection of sets A1 , A2 , . . . , An , whose union is A. Prove the following identity
holds for all objects x:
Yn
(σA (x) − σAi (x)) = 0.
i=1

(c) Using (a), (b), and the relationship between σS (x) and |S|, prove the Principle of Inclusion
and Exclusion.

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Algebra 2.5 AwesomeMath Summer Program 2024

2 Sources
1. 2013 HMMT February Algebra #5

2. 2010 PUMaC Algebra A #5

3. 2013 CHMMC Tiebraker #4

4. 1985 NYSML Individual #4

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5. 2016 HMMT November Team #3

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7. 2009 PUMaC Algebra A #2

8. 2018 SMT Algebra #10

9. 2010 CHMMC Winter Tiebreaker #1

10. 2010 AMC 12B #24

11. 1996 Canada #1

12. 2017 PUMaC Algebra A #6

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13. 2009 APMO #2

14. 2015 HMMT February Algebra #5

15. 2016 CMIMC Algebra #8

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16. 1988 IMO #4

17. 1994 Baltic Way #4

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18. 2021 Purple Comet High School #28

19. 2014 HMMT February Guts #12

20. 1984 AIME #15

21. 2014 USAMO #1

22. 1978 Canada #3

23. 2015 AIME I #13