ABJeIIy

444 Questions
17 Answers

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Algebra

The sum
6\left( 1\cdot1 + 2\cdot2 + \dots + n(n) \right)
is equal to a polynomial f(n) for all n \ge 1.
Write f(n) as a polynomial with terms in descending order of n.

ABJeIIy Aug 30, 2024

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+1694

Geometry

Let $PQRSTUVW$ be a rectangular prism, as shown. The area of face $TUVW$ is $10,$ the area of face $TUQP$ is $10,$ and the area of face $TPSW$ is $10.$ Find the volume of the rectangular prism.

ABJeIIy Aug 24, 2024

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+1694

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 90^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle XZW$, in degrees.

ABJeIIy Aug 23, 2024

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+1694

Geometry

Let $P$ be a point inside rectangle $ABCD$. If $AP = 3$, $BP = 3$, and $CP = 3$, then find $DP$.

ABJeIIy Aug 23, 2024

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+1694

Algebra

For a certain value of $k$, the system
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y
has no solutions. What is this value of $k$?

ABJeIIy Aug 19, 2024

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+1694

Algebra

Find the ordered quintuplet (a,b,c,d,e) that satisfies the system of equations
a+2b+3c+4d+5e=47
2a+3b+4c+5d=15
3a+4b+5c=34 read more ..

ABJeIIy Aug 19, 2024

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+1694

Algebra

Let x and y be real numbers. If x and y satisfy x^2+y^2=4x+16y+15, then find the largest possible value of x. Give your answer in exact form using radicals, simplified as far as possible.

ABJeIIy Aug 19, 2024

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+1694

Algebra

Let $a$ and $b$ be the roots of $x^2 + 7x - 4 = -x^2 + 9x + 6$. Find $(a + 3) + (b + 3).$

ABJeIIy Aug 17, 2024

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+1694

Algebra

One of the five quadratics below has a repeated root. (The other four have distinct roots.) What is the repeated root?
x^2 - 2x + 1
x^2 - 4x - 4
x^2 + 6x - 9 read more ..

ABJeIIy Aug 17, 2024

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+1694

Algebra

Let x and y be nonnegative real numbers. if x^2+5y^2=30, what is the max value of x + y?

ABJeIIy Aug 15, 2024

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+1694

Geometry

In triangle ABC, the angle bisector of angle BAC meets BC at D, such that AD = AB. Line segment AD is extended to E, such that angle DBE = angle BAD = 17 degrees. Find angle ABD.

ABJeIIy Aug 14, 2024

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+1694

Geometry

In the diagram below, D is a point on segment CE such that segments AD and BE are parellel and point A is not on segment BC. Line segments AD and BC intersect at point P.
If angle CAD = 67 degrees, then what is angle read more ..

ABJeIIy Aug 14, 2024

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+1694

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

ABJeIIy Aug 13, 2024

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+1694

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

ABJeIIy Aug 13, 2024