Stanry

106 Questions
12 Answers

0

4

0

+330

Number Theory

Which of the residues $0,$ $1,$ $2,$ $3,$ $4$ satisfy the congruence $3x \equiv 0 \pmod{5}?$
Give your answer as a list, separated by commas, in order from least to greatest.

Stanry Oct 29, 2024

0

6

0

+330

Number Theory

Which of the residues $0,$ $1,$ $2,$ $3,$ $4,$ $5$ satisfy the congruence $x^2 \equiv 0 \pmod{6}?$
Give your answer as a list, separated by commas, in order from least to greatest.

Stanry Oct 29, 2024

0

5

0

+330

Number Theory

Let $a$ be an integer such that $a \equiv 5 \pmod{7}$. Find the value of $a + 1 \pmod{7}$. Express your answer as a residue between $0$ and the modulus.

Stanry Oct 29, 2024

0

7

0

+330

Number Theory

Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$. If $a \neq 0,$ then find the value of $a$.

Stanry Oct 29, 2024

-1

6

0

+330

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 $a$.

Stanry Oct 23, 2024

0

3

1

+330

Number Theory

Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$.

Stanry Oct 23, 2024

0

4

0

+330

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 ..

Stanry Oct 23, 2024

0

6

1

+330

Number Theory

You have a total supply of $1000$ pieces of candy, and an empty vat. You also have a machine that can add exactly $52$ pieces of candy per scoop to the vat, and another machine that can remove exactly $39$ pieces of candy with a different scoop from read more ..

Stanry Oct 23, 2024

0

5

0

+330

Counting

In how many ways can you distribute $8$ indistinguishable balls among $5$ distinguishable boxes, if at least three of the boxes must be empty?

Stanry Oct 20, 2024

0

8

0

+330

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?

Stanry Oct 20, 2024

0

6

1

+330

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 ..

Stanry Oct 19, 2024

0

7

1

+330

Counting

A standard six-sided die is rolled $8$ times. You are told that among the rolls, there was two $1$'s, two $2$'s, two $3$'s, and two $4$s. How many possible sequences of rolls could there have been? (For example, $2,$ $1,$ $3,$ $4,$ $2,$ read more ..

Stanry Oct 19, 2024

0

5

0

+330

Counting

We call a number cozy if every digit in the number is either a $3$ or next to a $3.$ For example, the numbers $333,$ $83,$ $303,$ and $3773$ are all cozy, but the numbers $32423,$ $786,$ $340,$ and $3999$ are not cozy.
How many read more ..

Stanry Oct 19, 2024

0

11

0

+330

Geometry

Point $A$ is reflected over the line shown below to point $B$. Find the coordinates of $B$.
Write your answer as an ordered pair $(x,y)$.
The line is y = x + 3, and A = (7,1).

Stanry Sep 30, 2024

0

10

0

+330

Geometry

Let a be a real number such that 0 < a < \frac{\pi}{2}. Show that \left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 2*sin(a)^4*cos(a)^4.

Stanry Sep 30, 2024

0

10

0

+330

Geometry

Let $ABC$ be a triangle. Let $D$ be a point on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 30^\circ$, $\angle C = 90^\circ$, and $AC = 12$, then find the area of triangle $ABD$. Give your answer in exact read more ..

Stanry Sep 30, 2024