magenta

158 Questions
19 Answers

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Geometry

In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

magenta 10 hours ago

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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 not sit next to each other in the same row?

magenta 4 sept 2024

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Geometry

Point $P$ is on side $\overline{AC}$ of triangle $ABC$ such that $\angle APB =\angle PBA$, and $\angle ABC - \angle ACB = 24^\circ$. Find $\angle PBC$ in degrees.

magenta 3 sept 2024

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Counting

Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are together? (Two seatings are considered the same if one can be rotated to form the other.)

magenta 30 ago 2024

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Geometry

A rectangular prism has a length of $3$, a width of $5$, and a height of $11$. What is the distance between two opposite corners of the prism?

magenta 24 ago 2024

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Counting

A coin is tossed $12$ times. How many different sequences of tosses could there have been, if the number of heads was at least $5?$

magenta 18 ago 2024

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Counting

In how many ways can we distribute 13 pieces of identical candy to 3 kids, if the two youngest kids are twins and insist on receiving an equal number of pieces?

magenta 18 ago 2024

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Geometry

Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

magenta 15 ago 2024

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Number Theory

Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{7}.

magenta 14 ago 2024

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Number Theory

When $N$ is divided by $10$, the remainder is $a$. When $N$ is divided by $2$, the remainder is $b$. What is $N$ modulo $20$, in terms of $a$ and $b$?

magenta 14 ago 2024

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Geometry

Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at $P$. If $AP = 3 \cdot PB$, then what is $AB$?

magenta 10 ago 2024

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Geometry

A semicircle is inscribed in triangle $XYZ$ so that its diameter lies on $\overline{YZ}$, 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.

magenta 10 ago 2024

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Algebra

I drove to the beach at a rate of 40 miles per hour. If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later. How many miles did I drive?

magenta 10 ago 2024

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Algebra

Find the real number k such that f(x) = f(k - x) for all real numbers x, where
f(x) = 3x^2 - 4x + 2x^2 + 15x.

magenta 10 ago 2024

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Algebra

Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b \cdot g(x)$, where $f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5$ and $g(x) = x^15 - 3x^14 - 2x^13 - 6x^12 + 14x - 10$?

magenta 10 ago 2024

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Coordinate question

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

magenta 9 ago 2024

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Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ lee mas ..

magenta 6 ago 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 6 from one wall, and a distance of 6 from the other wall. Find the radius of the lee mas ..

magenta 6 ago 2024