nathanl6656

83 Questions
16 Answers

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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 $AB.$ read more ..

nathanl6656 Dec 13, 2024

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Geometry

Lines $XQ$ and $XR$ are tangent to a circle, as shown below. If $\angle XTU = 47^\circ$ and $\angle QAU = 65^\circ,$ then find $\angle QXR,$ in degrees.

nathanl6656 Dec 13, 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 read more ..

nathanl6656 Dec 13, 2024

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Geometry

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant read more ..

nathanl6656 Dec 13, 2024

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Geometry

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

nathanl6656 Dec 12, 2024

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Geometry

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 $a + b$.

nathanl6656 Dec 12, 2024

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Geometry

In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$

nathanl6656 Dec 12, 2024

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Counting

You are given the 4 x 4 grid below.
(a) Find the number of ways of placing 8 counters in the squares (at most one counter per square), so that each row contains exactly two counters.
(b) Find the number of ways read more ..

nathanl6656 Dec 5, 2024

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Counting

In how many ways can we distribute 13 pieces of identical candy to 5 kids, if the two youngest kids are twins and insist on receiving at least four pieces each?

nathanl6656 Dec 5, 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?

nathanl6656 Dec 5, 2024

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6

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Polygons

A regular a-gon, a regular b-gon, a regular c-gon, and a regular d-gon fit perfectly around a point. What is the largest possible value of c?

nathanl6656 Oct 30, 2024

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Number Theory

In the triangular array below, n different prime numbers p_1, p_2, \dots, p_n are written in the bottom row. Then in each box, we write the product of the two numbers below it. The number $K = p_1^{\alpha_1} p_2^{\alpha_2} \dotsm p_n^{\alpha_n}$ read more ..

nathanl6656 Oct 30, 2024

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Number Theory

Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{3}, \\
N &\equiv 2 \pmod{7}, \\
N &\equiv 2 \pmod{10}.

nathanl6656 Oct 30, 2024