Jasmo

109 Questions
5 Answers

0

8

2

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Counting

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

Jasmo May 10, 2025

0

32

0

+456

Algebra

Suppose the real numbers $a$, $b$, $x$, and $y$ satisfy the equations
ax + by = 3,
ax^2 + by^2 = 5,
ax^3 + by^3 = 17,
ax^4 + by^4 = 23.<#> Evaluate ax^5 + by^5.

Jasmo Mar 2, 2025

0

28

0

+456

Algebra

Let a, b, c, and d be distinct real numbers such that
a = \sqrt{4 + \sqrt{5 + a}},
b = \sqrt{4 - \sqrt{7 + b}},
c = \sqrt{4 + \sqrt{9 - c}},
d = \sqrt{4 - \sqrt{11 - d}}.
Compute abcd.

Jasmo Mar 2, 2025

0

29

0

+456

Algebra

If a_1, a_2, ..., a_19 satisfy
a_1 + a_2 + a_3 + a_4 = 1,
a_2 + a_3 + a_4 + a_5 = 2,
a_3 + a_4 + a_5 + a_6 = 3,
...
a_{16} + a_{17} + a_{18} + a_{19} = 16,
a_{17} + a_{18} + a_{19} + read more ..

Jasmo Mar 1, 2025

0

29

0

+456

Algebra

Let a = 3(x - y), b = 3(y - z), and c = 3(z - x), where x, y, z are real numbers, and assume ab + ac + bc \ne 0. Compute
(a^3 + b^3 + c^3)/(ab + ac + bc).

Jasmo Mar 1, 2025

0

33

0

+456

Algebra

Express x^8 + x^4 y^4 - x^6 y^2 + x^3 y^5 - 4xy^7 - y^8 as the product of two polynomials of degree 2, and a polynomial of degree 4.

Jasmo Feb 27, 2025

0

21

1

+456

Algebra

Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.

Jasmo Feb 27, 2025

0

24

0

+456

Algebra

If (x,y) satisfies the simultaneous equations
3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.
x^2 - y^2 = 7 + 4xy
where x and y may be complex numbers, determine all possible values of y^2.

Jasmo Feb 26, 2025

0

17

0

+456

Algebra

Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.

Jasmo Feb 26, 2025

0

22

0

+456

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

Jasmo Feb 25, 2025

0

28

0

+456

Algebra

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

Jasmo Feb 25, 2025

0

32

0

+456

Algebra

Factor x^2 - 2x + 7 - y^2 - 10yz - 13z^2 as the product of two polynomials of degree 1.

Jasmo Feb 25, 2025

0

25

2

+456

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 7n - 3mn + 15m + 8n + ___

Jasmo Feb 25, 2025

0

15

1

+456

Algebra

Suppose r and s are the values of x that satisfy the equation
x^2 - 2mx + (m^2 - 6m + 11) = 0
for some real number m. Find the minimum real value of (r - s)^2.

Jasmo Feb 24, 2025

0

16

1

+456

Algebra

Let r, s, and t be solutions of the equation x^3 + 2x^2 - 5x + 15 = 0. Compute
\frac{1}{r - 2s - 2t} + \frac{1}{s - 2r - 2t} + \frac{1}{t - 2r - 2s}

Jasmo Feb 24, 2025

0

17

3

+456

Algebra

Let r, s, and t be solutions of the equation x^3 - 4x^2 - 7x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}

Jasmo Feb 24, 2025

0

28

0

+456

Algebra

Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

Jasmo Feb 24, 2025

+1

15

2

+456

Algebra

Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2 qrs + pq^2 rs + pqr^2 s + pqrs^2.

Jasmo Feb 23, 2025

+1

19

2

+456

Algebra

Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2 + q^2 + r^2 + s^2.

Jasmo Feb 23, 2025

+456

-1

The soltuions are -5, 2, 8, and 15.

JasmoFeb 27, 2025

+456

0

The answer is (b,c,d) = (-2,8,15).

JasmoFeb 22, 2025

+456

0

The answer is (b,c) = (5,-15).

JasmoFeb 22, 2025

+456

0

The roots of f(x) are -1, -8, and -10.

JasmoFeb 22, 2025

+456

0

The possible values of m are -5, 1, 2, and 13.

JasmoFeb 22, 2025