All Questions ⁺⁰235535 Questions

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Geometry

Franklin the Fly lives at one corner of a cube of side length 1. He doesn't like to wander too far from home; in fact, Franklin is only willing to travel to a point on the cube if he can reach it by crawling a distance of at most \sqrt{5} along the read more ..

onyuIee Oct 30, 2024

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+430

Geometry

In triangle ABC, D is a point on side \overline{BC} such that $DC = AB.$ Let $M$ be the midpoint of $\overline{BD},$ and let $N$ be the midpoint of $\overline{AC}.$ Let $\theta = \angle NAC.$ Then what is \angle ABC in terms of \theta read more ..

onyuIee Oct 30, 2024

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+207

Geometry

In triangle ABO, M is the midpoint of \overline{AB}. Points C, D, and N lie on line segments $\overline{AO},$ $\overline{BO},$ and $\overline{MO}$ extended, respectively, so that $DO = BO,$ $NO = MO,$ and $CO = AO.$ If $K$ is the area of triangle read more ..

BRAlNBOLT Oct 30, 2024

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+207

Geometry

In the diagram below, ABCD is a square, DE = EF = FG = 2, GH = 1, HB = 5, and \angle DEF = \angle EFG = \angle FGH = \angle GHB = 60^\circ. Find the area of the shaded region.

BRAlNBOLT Oct 30, 2024

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+207

Geometry

Unit square ABCD is divided into two triangles, which are congruent to each other, and two trapezoids, which are also congruent to each other. These pieces are then re-assembled into rectangle $PQRS,$ as shown below. Find the perimeter of rectangle read more ..

BRAlNBOLT Oct 30, 2024

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+1218

Algebra

For a positive integer n, let
a_n = 1 + \sqrt{\frac{1}{n}} - \sqrt{\frac{1}{n + 1}

Compute the product a_1 a_2 a_3 \dotsm a_{99}.

learnmgcat Oct 30, 2024

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+1218

Algebra

Ten positive integers (not necessarily distinct) are arranged in a circle. An integer is colored red if it is equal to the average of its two neighbors. What is the largest possible number of red numbers?

learnmgcat Oct 30, 2024

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+1218

Algebra

Consider the set of all fractions of the form \frac{2^m}{3^n}, where m and n are integers such that 0 \le m \le n \le 3. For example, \frac{1}{9} and \frac{8}{27} would be included in this set, but not \frac{32}{81}. What is the sum of all the read more ..

learnmgcat Oct 30, 2024

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+1218

Algebra

For certain real numbers a and b, the function f satisfies f(x) = ax + b + cx^2 and f(bx + a) = x + cx^2 for all real numbers x. Find a + b.

learnmgcat Oct 30, 2024

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+306

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

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+306

Number Theory

Find the smallest positive integer such that
N &\equiv 4 \pmod{6}, \\
N &\equiv 4 \pmod{10}, \\
N &\equiv 4 \pmod{15}, \\
N &\equiv 4 \pmod{24}.

nathanl6656 Oct 30, 2024

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+297

Geometry

In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 15$, $BY = 20$, $\angle ACB= 60^\circ$, and $CY = 3 \cdot CX$, then find $AB$.

rtsdylifdts Oct 30, 2024

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+297

Geometry

Points $A$ and $B$ are on side $\overline{YZ}$ of rectangle $WXYZ$ such that $\overline{WA}$ and $\overline{WB}$ trisect $\angle ZWX$. If $WX = 2$ and $XY = 3$, then what is the area of rectangle $WXYZ$?

rtsdylifdts Oct 30, 2024

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+297

Geometry

Let $G$ be the center of equilateral triangle $XYZ.$ A dilation centered at $G$ with scale factor $-1$ is applied to triangle $XYZ,$ to obtain triangle $X'Y'Z'.$ Let $A$ be the area of the region that is contained in both triangles $XYZ$ and $X'Y'Z'.$ Find read more ..

rtsdylifdts Oct 30, 2024

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+297

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.

rtsdylifdts Oct 30, 2024

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+952

Number Theory

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

Hi6942O Oct 30, 2024

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+952

Number Theory

The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
101010010101000101A10001B001010C100011_2.
Compute $A,$ $B,$ and $C$.

Hi6942O Oct 30, 2024

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+952

Number Theory

A school orders $99$ textbooks, all for the same price. When the bill for the total order comes, the first and last digits are obscured. What are the missing digits?
_18,486.7_

Hi6942O Oct 30, 2024

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+36

Number Theory

When a prime is divided by 60 the remainder is a composite number. Find the sum of the digits of this composite number.

HelpIneedHelp Oct 30, 2024

Oct 29, 2024

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+1427

Number Theory

The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$. Find the remainder when $F_{1000}$ is divided by $17$.

blackpanther Oct 29, 2024

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+1427

Number Theory

How many of the 1000 smallest positive integers are congruent to 7 modulo 29?

blackpanther Oct 29, 2024

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+4

help geometry!!!

As shown in the diagram, points B and D are on different sides of line AC. We know that Angle B=2*Angle D=60 degrees and that AC=4sqrt(3). What is the distance between the circumcenters of Triangle ABC and Triangle ADC?

evan10203040 Oct 29, 2024

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+936

Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

Rangcr897 Oct 29, 2024

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