Jasmo

109 Questions
5 Answers

0

5

2

+456

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 10 may 2025

0

25

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 2 mar 2025

0

23

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 2 mar 2025

0

25

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} + lee mas ..

Jasmo 1 mar 2025

0

25

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 1 mar 2025

0

29

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 27 feb 2025

0

19

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 27 feb 2025

0

21

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 26 feb 2025

0

13

0

+456

Algebra

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

Jasmo 26 feb 2025

0

19

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 25 feb 2025

0

22

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 25 feb 2025

0

25

0

+456

Algebra

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

Jasmo 25 feb 2025

0

20

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 25 feb 2025

0

14

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 24 feb 2025

0

12

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 24 feb 2025

0

12

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 24 feb 2025

0

22

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 24 feb 2025

+1

13

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 23 feb 2025

+1

16

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 23 feb 2025

+456

-1

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

Jasmo27 feb 2025

+456

0

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

Jasmo22 feb 2025

+456

0

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

Jasmo22 feb 2025

+456

0

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

Jasmo22 feb 2025

+456

0

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

Jasmo22 feb 2025