All Questions ⁺⁰236048 Questions

0

5

1

+781

Algebra

Let a and b be integers such that the polynomial
x^4 + ax^3 + bx^2 + ax + 1 = 0
has four distinct positive real roots. Find the smallest possible value of a + b.

booboo44 Oct 29, 2024

0

3

1

+781

Algebra

Let $d$ be a function taking the positive integers to the nonnegative integers such that $d(p) = 1$ for any prime $p,$ and
d(ab) = b \cdot d(a) + a \cdot d(b)
for all positive integers $a$ and $b.$ Find the number of positive integers read more ..

booboo44 Oct 29, 2024

0

6

1

+781

Algebra

Let r, s, and t be the real roots of
(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = 0.
Compute r^3 + s^3 + t^3.

booboo44 Oct 29, 2024

0

3

0

+379

Algebra

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant read more ..

MeIdHunter Oct 29, 2024

0

7

0

+379

Algebra

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

MeIdHunter Oct 29, 2024

0

8

0

+379

Algebra

Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.

MeIdHunter Oct 29, 2024

0

8

0

+310

Number Theory

A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are 2, $F$, and 1. If the integer is a multiple of $19_{10}$, what is the units digit?

MEMEG0D Oct 29, 2024

0

3

1

+310

Number Theory

The numbers $24^2 = 576$ and $56^2 = 3136$ are examples of perfect squares that have a units digits of $6.$
If the units digit of a perfect square is $5,$ then what are the possible values of the tens digit?

MEMEG0D Oct 29, 2024

0

9

0

+310

Number Theory

Which of the residues 0, 1, 2, ..., 11 satisfy the congruence 3x = 1 mod 12?

MEMEG0D Oct 29, 2024

0

9

0

+310

Number Theory

Which of the residues 0, 1, 2, 3, 4 satisfy the congruence x^5 = 0 mod 5?

MEMEG0D Oct 29, 2024

0

5

0

+862

Algebra

Let $x$ and $y$ be real numbers such that $x^3 + 3xy^2 = 679$ and $x^3 - 3xy^2 = 615.$ Find $x - y.$

magenta Oct 29, 2024

0

7

0

+862

Algebra

When the same constant is added to the numbers 60, 120, and 160, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?

magenta Oct 29, 2024

0

4

0

+862

Algebra

Let a_1, a_2, a_3, \dots, a_{10}, a_{11}, a_{12} be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.

magenta Oct 29, 2024

0

4

0

+862

Algebra

Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 20, Shirley runs screaming from the room. What number does Laverne say that sends read more ..

magenta Oct 29, 2024

0

5

2

+4

Any idea?

In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.

Causeveras39 Oct 29, 2024

0

6

0

+781

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

booboo44 Oct 29, 2024

0

5

0

+781

Algebra

Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.

booboo44 Oct 29, 2024

0

3

0

+781

Algebra

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

booboo44 Oct 29, 2024

0

8

0

+781

Algebra

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

booboo44 Oct 29, 2024

0

4

0

+801

Number Theory

In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at least 5?

gnistory Oct 29, 2024

0

6

0

+801

Number Theory

Find the number of $7$-digit numbers, where the sum of the digits is divisible by $11.$

gnistory Oct 29, 2024

0

7

0

+801

Number Theory

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

gnistory Oct 29, 2024

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