All Questions ⁺⁰236050 Questions

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Counting

Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely. What is the probability that Dhruv gets at least $6$ chocolates?

kittykat Nov 18, 2024

Nov 17, 2024

0

6

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+976

Geometry

Point D is the midpoint of median AM of triangle ABC. Point E is the midpoint of AB, and point T is the intersection of BD and ME. Find the area of triangle BEC if [ABC] = 14.

Hi6942O Nov 17, 2024

0

3

1

+976

Geometry

In triangle $ABC$, $\angle ABC = 90^\circ$, and $D$ is on side $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. If $AB = 4,$ $BC = 3$, and $AC = 5,$ then find the area of $\triangle ADC$. Round your answer to the nearest integer.

Hi6942O Nov 17, 2024

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2

1

+976

Geometry

In triangle $ABC$, let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. If $BC = 20$ and $\angle C = 15^\circ$, then find the length of $BE$.

Hi6942O Nov 17, 2024

Nov 1, 2024

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7

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+1839

Algebra

The function
f(x) = \frac{cx}{7x - 3}
satisfies f(f(x))=0 for all real numbers $x\neq \frac{3}{7}$. Find c.

ABJeIIy Nov 1, 2024

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7

0

+1839

Algebra

Let $f(x)$ and $g(x)$ be functions. Find $c$ if
(f \circ g)(x) = (g \circ f)(x)
for all $x$, where $f(x) = 3x - 4$ and $g(x) = 5x + c$.

ABJeIIy Nov 1, 2024

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4

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+1839

Algebra

Find the real number $k$ such that $f(x) = f(k - x)$ for all real numbers $x$, where
f(x) = 3x^2 - 4x + x^2 + 17x.

ABJeIIy Nov 1, 2024

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7

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+1839

Algebra

Let $f$ be a function such that
f(x+y) = x + f(y) + f(x) - y
for any two real numbers $x$ and $y$. If $f(0) = -5$, then what is $f(17)?$

ABJeIIy Nov 1, 2024

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7

0

+1839

Algebra

Find the value of a for which there is exactly one real value of x such that f(x) = a, where
f(x) = x^2 + 4x - 31 + 16x + 18.

ABJeIIy Nov 1, 2024

0

7

0

+1839

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $x^2 + 5y^2 = 30$, then find the maximum value of $x + y$.

ABJeIIy Nov 1, 2024

0

5

1

+1839

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $xy = \frac{4}{3}$, then find the minimum value of $2x + y^6$.

ABJeIIy Nov 1, 2024

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4

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+1557

Number Theory

Find the number of bases $b \ge 2$ such that $100_b - 1_b$ is prime.

parmen Nov 1, 2024

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4

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+1557

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?

parmen Nov 1, 2024

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4

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+1557

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 $\frac{N}{24}?$

parmen Nov 1, 2024

Oct 31, 2024

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8

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+551

Number Theory

When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.

When read more ..

Pythagorearn Oct 31, 2024

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+551

Number Theory

Find the number of bases b \ge 2 such that 100_b - 1_b is prime.

Pythagorearn Oct 31, 2024

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5

0

+551

Number Theory

How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?

Pythagorearn Oct 31, 2024

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7

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+551

Number Theory

A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?

Pythagorearn Oct 31, 2024

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+551

Number Theory

The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$
What is the smallest positive integer that has exactly $2$ perfect square divisors?

Pythagorearn Oct 31, 2024

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5

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+551

Number Theory

Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{117}$.
Be sure to include complete explanations with your answer, using complete sentences. Imagine you were going to show your solution to a classmate, read more ..

Pythagorearn Oct 31, 2024

0

7

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+551

Number Theory

Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decim read more ..

Pythagorearn Oct 31, 2024

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6

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+4

. A helicopter flies on a bearing of from A to B, where . It then flies on a bearing of from B to C, where C is due south of A. Find the

. A helicopter flies on a bearing of from A to B, where . It then flies on a bearing of from B to C, where C is due south of A. Find the distance of C from A giving your answer to the nearest KM.

veeno Oct 31, 2024

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6

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+297

Number Theory

A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. How many prime numbers are divisors of $n?$

rtsdylifdts Oct 31, 2024

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6

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+297

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest number of primes that could divide a terrific positive integer?

rtsdylifdts Oct 31, 2024

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