LiIIiam0216

312 Questions
16 Answers

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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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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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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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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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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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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

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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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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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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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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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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