ChiIIBill

115 Questions
12 Answers

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Geometry

Triangle ABC has a right angle at B. Point D lies on side \overline{AC} such that CD = 6. The circle with diameter $\overline{CD}$ intersects $\overline{AB}$ at two distinct points, $E$ and $F,$ with $AE < AF.$ If AE = 6 and DE = 4, read more ..

ChiIIBill Oct 29, 2024

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Geometry

In right triangle ABC, \angle ACB = 90^\circ, and the legs are a = BC and b = AC. Let $S$ and $T$ be points on $\overline{AC}$ and $\overline{BC},$ respectively. Let $R$ be the foot of the altitude from $S$ to $\overline{AB},$ and let $U$ be read more ..

ChiIIBill Oct 29, 2024

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Geometry

In tetrahedron ABCD, AB = AC = AD = 12 and BC = BD = CD = 12. There is a sphere that is tangent to all six edges of the tetrahedron. Find the radius of this sphere.

ChiIIBill Oct 29, 2024

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73

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Algebra

Let a and b be real numbers such that a^3 + 3ab^2 = 679 and 3a^3 - ab^2 = 615. Find a - b.

ChiIIBill Sep 3, 2024

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Algebra

Let $m$ be a real number. If the quadratic equation $x^2+mx+4 = 2x^2 + 17x + 8$ has two distinct real roots, then what are the possible values of $m$? Express your answer in interval notation.

ChiIIBill Aug 13, 2024

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help algebra

If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?

ChiIIBill Aug 13, 2024

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54

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Number Theory

What is 100000_{16} * E21A8_{16} in base 16?

ChiIIBill Aug 7, 2024

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56

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Number Theory

Let N be a positive integer. The number N has three digits when expressed in base 7. When the number N is expressed in base 18, it has the same three digits, in reverse order. What is N? (Express your answer in decimal read more ..

ChiIIBill Aug 7, 2024

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Number Theory

What is the largest positive integer $n$ such that $1457$, $1797$, $709$, $15$, $24$, $197$, $428$ all leave the same remainder when divided by $n$?

ChiIIBill Jul 23, 2024

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59

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Number Theory

If n \equiv 43 \pmod{165}, then what is the residue of $n$ modulo 2?

ChiIIBill Jul 23, 2024

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64

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Number Theory

A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are $7$, $D$, and $3$. If the integer is a multiple of $25_{10}$, what is the units digit?

ChiIIBill Jul 23, 2024

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Number Theory

How many lattice points (points with integer coordinates) are on the line segment whose endpoints are $(3,17)$ and $(82,150)?$ (Include both endpoints in your count.)

NotThatSmart ●

ChiIIBill Jul 23, 2024

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Algebra

Let a1, a2, a3, ..., a10 be an arithmetic series. If a1 + a3 = -2 and a2 + a4 + a6 = 1, then find a1.

NotThatSmart ●

ChiIIBill Jul 23, 2024

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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 100, Shirley runs screaming from the room. What number does Laverne say that sends read more ..

NotThatSmart ●

ChiIIBill Jul 23, 2024

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57

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Algebra

Find the ordered pair that satisfies the system
\dfrac{s}{2} + 5t &= 3 - 3s + t,\\
3t - 6s &= 9 + 5s.

ChiIIBill Jul 23, 2024