All Questions ⁺⁰234919 Questions

0

15

0

+892

Number Theory

How many positive four-digit integers $N$ have the property that the three-digit number obtained by removing the leftmost digit is equal to $\frac{N}{24}?$

Rangcr897 Jul 6, 2024

0

9

0

+892

Number Theory

The product of a set of distinct positive integers is $630$. If one of the numbers is 42, what are the other two numbers?

Rangcr897 Jul 6, 2024

Jul 5, 2024

0

5

0

+228

Number Theory

Find the largest positive integer n such that
\frac{(n + 1)^2}{n + 2}
is an integer.

nathanl6656 Jul 5, 2024

0

11

0

+228

Number Theory

Find the largest positive integer n such that
\frac{(n + 1)^2}{n + 2}
is an integer.

nathanl6656 Jul 5, 2024

0

11

0

+228

Number Theory

Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and
a + 3n = b - 2n = \frac{c}{5n},
compute the value of $a$.

nathanl6656 Jul 5, 2024

0

9

0

+228

Number Theory

How many positive four-digit integers $N$ have the property that the three-digit number obtained by removing the leftmost digit is equal to $\frac{N}{10}?$

nathanl6656 Jul 5, 2024

0

6

0

+228

Counting

Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?
A
RRR
CCCCC read more ..

nathanl6656 Jul 5, 2024

0

10

0

+228

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

nathanl6656 Jul 5, 2024

0

7

0

+228

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

nathanl6656 Jul 5, 2024

0

12

0

+228

Number Theory

Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?

nathanl6656 Jul 5, 2024

0

7

0

+228

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

nathanl6656 Jul 5, 2024

0

12

0

+790

Geometry

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

eramsby1O1O Jul 5, 2024

0

3

1

+790

Geometry

Points $M$, $N$, and $O$ are the midpoints of sides $\overline{KL}$, $\overline{LJ}$, and $\overline{JK}$, respectively, of triangle $JKL$. Points $P$, $Q$, and $R$ are the midpoints of $\overline{NO}$, $\overline{OM}$, and $\overline{MN}$, respectively. read more ..

eramsby1O1O Jul 5, 2024

0

11

1

+790

Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $AMC$ if $[ABC] = 14$. read more ..

eramsby1O1O Jul 5, 2024

0

9

0

+1724

help with geo

In trapezoid PQRS, \overline{PQ} \parallel \overline{RS}. Let X be the intersection of diagonals \overline{PR} and \overline{QS}. The area of triangle PRS is 27, and the area of triangle PQR is 75. Find the area of trapezoid PQRS read more ..

ABJeIIy Jul 5, 2024

0

7

0

+1724

help me plz

In trapezoid ABCD, \overline{AB} \parallel \overline{CD}. Find the area of the trapezoid.

ABJeIIy Jul 5, 2024

0

8

1

+1724

need help

Quadrilateral ABCD is a parallelogram. Let E be a point on \overline{AB}, and let F be the intersection of lines DE and BC. The area of triangle EBF is 1, and the area of triangle BCE is 2. Find the area of parallelogram ABCD.

ABJeIIy Jul 5, 2024

0

8

1

+1724

help me

In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.

ABJeIIy Jul 5, 2024

0

3

2

+31

Find , given that

Find , given that

AHGSIGMA Jul 5, 2024

0

8

0

+892

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 ..

Rangcr897 Jul 5, 2024

0

9

0

+892

Geometry

We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude to $\triangle ABC$. If $AC = 8,$ $BK = 2$, and $CK = 3,$ then what is $AB$?

Rangcr897 Jul 5, 2024

0

13

0

+892

Geometry

In triangle $ABC$, $AB = 10$ and $AC = 15$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 1:3$, then find $AD$.

Rangcr897 Jul 5, 2024

0

5

0

+892

Geometry

In triangle $ABC$, $AB = 7$, $AC = 17$, and the length of median $AM$ is $12$. Find the area of triangle $ABC$.

Rangcr897 Jul 5, 2024

0

8

0

+700

Geometry

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 14$, $AY = 14$, and $CY = 8$, find $BC$.

magenta Jul 5, 2024

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