All Questions ⁺⁰232936 Questions

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Coordinates

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 $a + b$.

eramsby1O1O May 19, 2024

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+1367

Geometry

Two circles are externally tangent at T. The line AB is a common external tangent to the two circles, and P is the foot of the altitude from T to line AB. Find the length AB.

parmen May 19, 2024

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+1367

Geometry

A semicircle is inscribed in triangle XYZ so that its diameter lies on YZ, and is tangent to the other two sides. If XY = 10, XZ = 10, and YZ = 10*sqrt(2), then find the area of the semicircle.

parmen May 19, 2024

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+16

Help pls, not urgent though

A mathematician works for t hours per day and solves p problems per hour, where t and p are positive integers. One day, the mathematician drinks some coffee and discovers that he can now solve 4p +2 problems per hour. In fact, he only works read more ..

jjtseng May 19, 2024

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+1857

Geometry

Chords UV, WX, and YZ of a circle are parallel. The distance between chords UV and WX is 1, and the distance between chords WX and YZ is also 1. If UV =6 and YZ = 4, then find WX.

LiIIiam0216 May 19, 2024

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5

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+1857

counting

In the Olympic women's skating competition, the gold medal goes to first place, silver to second, and bronze to third. If there are $3$ skaters, including $3$ Americans, in how many ways can the medals be awarded to three of the $3$ skaters if exactly read more ..

LiIIiam0216 May 19, 2024

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+1857

Counting

Dogs in the GoodDog Obedience School win a blue ribbon for learning how to sit, a green ribbon for learning how to roll over, and a white ribbon for learning how to stay. There are $100$ dogs in the school.
$62$ have blue ribbons, $55$ read more ..

LiIIiam0216 May 19, 2024

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+8

Corner of a rectangular piece of paper of width inches is folded over so that it coincides with point on the opposite side.

Corner of a rectangular piece of paper of width inches is folded over so that it coincides with point on the opposite side. If inches, find the length in inches of fold .

nidishmthpro May 19, 2024

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4

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+1857

Counting

Find the number of positive integers that satisfy both the following conditions:
- Each digit is a 1 or a 2 or a 3
- The sum of the digits is 5

LiIIiam0216 May 19, 2024

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+1857

Counting

In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?

LiIIiam0216 May 19, 2024

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+8

find the greatest common divisor of 957 and 1537

nidishmthpro May 19, 2024

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+1857

Geometry

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $CY = 6$, find $BZ$.

LiIIiam0216 May 19, 2024

May 18, 2024

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+1857

Algebra

The temperature of a point (x,y) in the plane is given by the expression x^2 + y^2 - 4x + 2y - 12x + 14y + 26. What is the temperature of the coldest point in the plane?

LiIIiam0216 May 18, 2024

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+1857

Algebra

Let x and y be nonnegative real numbers. If xy = \frac{2}{5}, then find the minimum value of 6x + \frac{3}{5y}.

LiIIiam0216 May 18, 2024

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+1857

Algebra

Let x and y be nonnegative real numbers. If x^2 + 3y^2 = 18, then find the maximum value of x + y.

LiIIiam0216 May 18, 2024

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+1857

Geometry

A circular table is pushed into a corner of the room, where two walls meet at a right angle. A point $P$ on the edge of the table (as shown below) has a distance of $10$ from one wall, and a distance of $13$ from the other wall. Find the radius read more ..

LiIIiam0216 May 18, 2024

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Geometry

The two circles below are externally tangent. A common external tangent intersects line $PQ$ at $R.$ Find $QR.$

LiIIiam0216 May 18, 2024

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+1857

Geometry

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $1$ and $BC = \sqrt{3}$, then find $AB$. read more ..

LiIIiam0216 May 18, 2024

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+1857

Geometry

In triangle PQR, M is the midpoint of PQ. Let X be the point on QR such that PX bisects angle QPR, and let the perpendicular bisector of PQ intersect AX at Y. If PQ = 36, PR = 22, QR = 26, and MY = 8, then find the area of triangle PQR

LiIIiam0216 May 18, 2024

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+1857

Geometry

A circle lies inside a quarter-circle, as shown below. The circle is tangent to side $\overline{AO}$ and arc $AB.$ Find the radius of the circle.

LiIIiam0216 May 18, 2024

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4

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+1857

Geometry

In triangle $PQR,$ let $X$ be the intersection of the angle bisector of $\angle P$ with side $\overline{QR}$, and let $Y$ be the foot of the perpendicular from $X$ to side $\overline{PR}$. If $PQ = 8,$ $QR = 5,$ and $PR = 1,$ then compute the length read more ..

LiIIiam0216 May 18, 2024

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Algebra

In a store window, there was a box of berries having a total weight of 200 kg. The berries were 98% water, by weight. After two days in the sun, the water content of the berries was only 95%, by weight. What was the total weight of the berries read more ..

LiIIiam0216 May 18, 2024

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+1857

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 $S_1 = \frac{1}{5}$ and $S_2 = \frac{1}{10},$ then find $S_{15}.$

LiIIiam0216 May 18, 2024

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Algebra

In class, we derived that
\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1}.
Fill in the blanks to make a true equation:
\frac{5x}{(x - 1)(x^2 + 2)(x + 7)^3)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 2} + \frac{D}{x + 7} read more ..

LiIIiam0216 May 18, 2024

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