Algebra 2.

5 AwesomeMath Summer Program 2024

Symmetric Polynomials & Newton Sums

1 Problem Set 4
Remember: These problems all involve symmetric polynomials, but only some of them will
involve Newton Sums. They are mixed together.
Problem 1: If the sum of two numbers is 1 and their product is 1, then find the sum of their
cubes.

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Problem 2: If (x, y) is a solution to the system

xy = 6 and x2 y + xy 2 + x + y = 63,

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find x2 + y 2 .

Problem 3: Find x2 + y 2 if x and y are positive integers such that

x+
eM
xy + x + y = 71
x2 y + xy 2 = 880.

Problem 4: Real numbers x and y satisfy x + y = 4 and x · y = −2. What is the value of
x3
y2
+
y3
x2
+ y?
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17
Problem 5: Positive real numbers x, y satisfy the equations x2 + y 2 = 1 and x4 + y 4 = 18 . Find
xy.

Problem 6: Suppose that the sum of the squares of two complex numbers x and y is 7 and the
sum of the cubes is 10. What is the largest real value that x + y can have?
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Problem 7: Let x, y, z be positive real numbers satisfying the simultaneous equations

x(y 2 + yz + z 2 ) = 3y + 10z
y(z 2 + zx + x2 ) = 21z + 24x
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z(x2 + xy + y 2 ) = 7x + 28y.

Find xy + yz + zx.
p
3
√ p
3
√
Problem 8: Compute 5 + 2 13 + 5 − 2 13

Problem 9: If a, b, c are non-zero real numbers such that

a+b−c a−b+c −a + b + c
= = ,
c b a
and
(a + b)(b + c)(c + a)
x= ,
abc
and x < 0, then x must be a constant value independent of a, b, and c. Find it.

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Algebra 2.5 AwesomeMath Summer Program 2024
Problem 10: Let a, b, c be the three roots of p(x) = x3 + x2 − 333x − 1001. Find a3 + b3 + c3 .

Problem 11: Let sk denote the sum of the kth powers of the roots of the polynomial x3 − 5x2 +
8x − 13. In particular, s0 = 3, s1 = 5, and s2 = 9. Let a, b, and c be real numbers such that
sk+1 = ask + bsk−1 + csk−2 for k = 2, 3, . . . . What is a + b + c?

20
Problem 12: Consider the polynomial x2017 + 2017x + 17 . Find the sum of the 2017th powers of
the roots of this polynomial.

Problem 13: (a) Given real numbers a, b, c, with a + b + c = 0, prove that a3 + b3 + c3 > 0 if
and only if a5 + b5 + c5 > 0.

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(b) Given real numbers a, b, c, d, with a + b + c + d = 0, prove that a3 + b3 + c3 + d3 > 0 if and
only if a5 + b5 + c5 + d5 > 0.

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Problem 14: A convex quadrilateral is determined by the points of intersection of the curves
x4 + y 4 = 100 and xy = 4; determine its area.

Problem 15: The equation


x−
√
3
 √
3

eM √
3
 1
13 x − 53 x − 103 =
3
has three distinct real solutions r, s, and t for x. Calculate the value of r3 + s3 + t3 .
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Problem 16: Let a, b, c, and d be real numbers that satisfy the system of equations

a + b = −3
ab + bc + ca = −4
abc + bcd + cda + dab = 14
abcd = 30.
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There exist relatively prime positive integers m and n such that

m
a2 + b2 + c2 + d2 = .
n
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Find m + n.

Problem 17: Let a, b, c, x, y, and z be complex numbers such that

b+c c+a a+b
a= , b= , c= .
x−2 y−2 z−2
If xy + yz + zx = 67 and x + y + z = 2010, find the value of xyz.

Problem 18: Consider the polynomials P (x) = x6 − x5 − x3 − x2 − x and Q(x) = x4 − x3 − x2 − 1.

Given that z1 , z2 , z3 , and z4 are the roots of Q(x) = 0, find P (z1 ) + P (z2 ) + P (z3 ) + P (z4 ).
p √
Problem 19: There are positive integers m and n such that m2 − n = 32 and 5
m+ n+
p √
5
m − n is a real root of the polynomial x5 − 10x3 + 20x − 40. Find m + n.

2
Algebra 2.5 AwesomeMath Summer Program 2024
Problem 20: Let x and y be real numbers satisfying x4 y 5 + y 4 x5 = 810 and x3 y 6 + y 3 x6 = 945.
Evaluate 2x3 + (xy)3 + 2y 3 .

Problem 21: How many ordered triples (x, y, z) of real numbers satisfy the system of equations

x2 + y 2 + z 2 = 9,
x4 + y 4 + z 4 = 33,
xyz = −4?

Problem 22: Suppose x, y and z are integers that satisfy the system of equations

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x2 y + y 2 z + z 2 x = 2186
xy 2 + yz 2 + zx2 = 2188.

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Evaluate x2 + y 2 + z 2

Problem 23: If −1 < x < 1 and −1 < y < 1, define the ”relativistic sum” x ⊕ y to be
x+y

What is the value of

v= √
7

7
x⊕y =

eM
17 − 1
17 + 1
.
1 + xy
.

The operation ⊕ is commutative and associative. Let v be the number

√
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v⊕v⊕v⊕v⊕v⊕v⊕v⊕v⊕v⊕v⊕v⊕v⊕v⊕v?

(In this expression, ⊕ appears 13 times.)

Problem 24: If x, y, z are real numbers satisfying relations

π
es

x+y+z =1 and arctan x + arctan y + arctan z = ,

4
prove that x2n+1 + y 2n+1 + z 2n+1 = 1 holds for all positive integers n.
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x+y
Problem 25: For all real numbers x, y define x ⋆ y = 1+xy . Evaluate the expression

(· · · (((2 ⋆ 3) ⋆ 4) ⋆ 5) ⋆ · · · ) ⋆ 1995.

Problem 26: A positive integer n and a real number a are given. Find all n-tuples (x1 , . . . , xn )
of real numbers that satisfy the system of equations
n
X
xki = ak for k = 1, 2, . . . , n.
i=1

Problem 27: Solve the system (

x+y+z =1
x5 + y 5 + z 5 = 1
in real numbers.

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Algebra 2.5 AwesomeMath Summer Program 2024
Problem 28: Let a, b, c, and d be real numbers such that a + b + c + d = a7 + b7 + c7 + d7 = 0.
Show that a(a + b)(a + c)(a + d) = 0.

Problem 29: Let p(x) be the polynomial (1 − x)a (1 − x2 )b (1 − x3 )c · · · (1 − x32 )k , where a, b, . . . , k

are integers. When expanded in powers of x, the coefficient of x1 is −2 and the coefficients of x2 ,
x3 , . . . , x32 are all zero. Find k.

Problem 30: Let a, b, c, d, e, f be positive integers and let S = a + b + c + d + e + f . Suppose

that the number S divides abc + def and ab + bc + ca − de − ef − df . Prove that S is composite.

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at
eM
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es
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