All Questions ⁺⁰237665 Questions

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65

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+1808

Rectangles

Let $ABCD$ and $BEDF$ be two $8 \times 9$ rectangles that overlap, as shown. Find the area of the overlap.

blackpanther Dec 20, 2023

-1

108

0

+1808

Triangle

In triangle $ABC$, let $M$ be the midpoint of $\overline{AB}$ and $N$ be on $\overline{BC}$ such that $\overline{AN}$ is an altitude of $\triangle ABC$. If $AB = AC = 4$ and $BC = 2$, and $\overline{AN}$ and $\overline{CM}$ intersect at $X$, then read more ..

blackpanther Dec 20, 2023

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82

2

+1808

Triangle

In the diagram below, we have $\angle ABC = \angle CAB = \angle DEB=\angle BDE.$ Find $AB$ and $CD$.

blackpanther Dec 20, 2023

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74

1

+1808

Square

In the diagram, square ABCD has side length 3. Find the area of triangle PDQ.

blackpanther Dec 20, 2023

-1

57

1

+1808

Isosceles triangles

$\triangle ABC$ is an isosceles triangle. $\angle C = 90^\circ.$ Find the sum of all possible values of $\angle B,$ in degrees.

blackpanther Dec 20, 2023

0

44

1

+1348

Franklin the Fly

Franklin the fly starts at the point $(0,0)$ in the coordinate plane. At each point, Franklin takes a step to the right, left, up, or down. After $2$ steps, how many different points could Franklin end up at?

sandwich Dec 20, 2023

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103

0

+1348

Choosing circles

Find the number of ways of choosing three circles below, so that no two circles are next to each other.

sandwich Dec 20, 2023

0

84

1

+1932

Polynomials

Find all real numbers a such that the roots of the polynomial
x^3 + x + a
form an arithmetic progression and are not all real.

tomtom Dec 20, 2023

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49

1

+1556

Problem with areas

In triangle $ABE,$ $C$ and $D$ are points on sides $\overline{BE}.$ If $BD = 8$, $CE = 24$, $[ABC] = 4$, and $[ADE] = 8$, then find $[ACD]$.

kittykat Dec 20, 2023

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73

1

+1556

Geometry

Let $ABCD$ be a square. Let $P$ be a point outside square $ABCD$ such that triangle $ABP$ is equilateral. Find $\angle BPD$ in degrees.

kittykat Dec 20, 2023

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68

1

+1556

Rectangle

In rectangle $ABCD$, as shown below, $DX = XC$. Find the number of degrees in $\angle XDY.$

kittykat Dec 20, 2023

0

60

1

+1556

triangle

In triangle $ABC$, $AB = AC$. Point $P$ lies on $\overline{AB}$ such that $CP = BC$. What is $\angle ACP$ (in degrees)?

kittykat Dec 20, 2023

Dec 19, 2023

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85

0

+1556

Altitudes

I am really stuck on this question.
Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 18,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$ read more ..

kittykat Dec 19, 2023

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81

1

+1556

Triangles

All the sides of a triangle are integers. If the perimeter of the triangle is $5,$ then how many different possible triangles are there? (Assume that the triangle is non-degenerate. Two triangles are considered the same if they are co read more ..

kittykat Dec 19, 2023

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57

2

+1556

Triangle question

The perimeter of a triangle is an integer. If one side of the triangle is $1$, then what is the smallest possible value of the perimeter? (Assume that the triangle is non-degenerate.)

kittykat Dec 19, 2023

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48

2

+1733

Math problem

My square patio is tiled with square tiles, all the same size. All the tiles are gray, except the tiles along the two diagonals, which are all yellow. (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between read more ..

parmen Dec 19, 2023

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45

1

+1733

Probability

A member of the junior varsity swim team will be elected as President. Each member is equally likely to be elected. The probability that a sophomore is elected is $4$ times the probability that a freshman is elected. Everyone on the read more ..

parmen Dec 19, 2023

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77

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+1733

help plz

Sam writes down the numbers $1,$ $2,$ $\dots,$ $315,$ $316,$ $\dots,$ $999$
Sam chooses one of the digits written down at random. What is the probability that Sam chooses a $2$?

parmen Dec 19, 2023

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62

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+1733

Help with problem

Sam writes down the numbers $1,$ $2,$ $\dots,$ $315,$ $316,$ $\dots$, $999$.
How many digits did Sam write, in total?

parmen Dec 19, 2023

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81

1

+1733

Need help

Joanna has seven beads that she wants to assemble into a bracelet. Five of the beads have the same color, and the other two all have different colors. In how many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical read more ..

parmen Dec 19, 2023

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67

0

+1776

Counting

We call a number cozy if every digit in the number is either even or next to an even digit. For example, a number is cozy if every digit in the number is either a 2 or next to a 2. For example, the number 222, 72, 45, and 810 and read more ..

bader Dec 19, 2023

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86

0

+1776

Counting

We call a number cozy if every digit in the number is either even or next to an even digit. For example, a number is cozy if every digit in the number is either a 2 or next to a 2. For example, the number 222, 72, 45, and 810 and read more ..

bader Dec 19, 2023

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82

0

+1776

Counting question

North High School has a math club and a science club. There are a students in the math club and b students who are members of both clubs. The total number of students is c. Find an expression for the total number of students read more ..

bader Dec 19, 2023

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87

0

+1776

Counting question

North High School has a math club and a science club. There are a students in the math club and b students who are members of both clubs. The total number of students is c. Find an expression for the total number of students read more ..

bader Dec 19, 2023

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