Practice

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2289

Compute $$\cos\frac{2\pi}{7} \cdot \cos \frac{4\pi}{7}\cdot \cos \frac{8\pi}{7} $$

2292

Compute $$\sin^2 10^\circ + \cos^2 40^\circ + \sin 10^\circ \cos 40^\circ$$

2334

Let $a$, $b$, $c$, $d$, and $e$ be five positive integers. If $ab+c=3115$, $c^2+d^2=e^2$, both $a$ and $c$ are prime numbers, $b$ is even and has $11$ divisors. Find these five numbers

2355

Compute the value of $\sin{18^\circ}$ using regular geometry.

2360

Let $k$ be a positive integer, show that $(4k+3)$ cannot be a square number.

2361

How many numbers in this series are squares? $$1, 14, 144, 1444, 14444, \cdots$$

2363

Solve the following equation in positive integers: $15x - 35y + 3 = z^2$

2365

Solve the following equation in positive integers: $3\times (5x + 1)=y^2$

2366

Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.

2389

Let both $A$ and $B$ be two-digit numbers, and their difference is $14$. If the last two digits of $A^2$ and $B^2$ are the same, what are all the possible values of $A$ and $B$.

2390

Find such a positive integer $n$ such that both $(n-100)$ and $(n-63)$ are square numbers.

2391

Find such a positive integer $n$ such that both $(n+23)$ and $(n-30)$ are square numbers.

2392

Find the smallest positive integer $n$ such that $\frac{12!}{n}$ is a square.

2414

Find all the integer solutions to the equation $xy - 10(x+ y)= 1$.

2415

Solve in integers the equation $x^2 - xy +2x -3 y = 0$

2440

In $\triangle{ABC}$, $AB=AC$. Extending $CA$ to an arbitrary point $P$. Extending $AB$ to point $Q$ such that $AP=BQ$. Let $O$ be the circumcenter of $\triangle{ABC}$. Show that $A$, $P$, $Q$, and $O$ concyclic.

2446

Let function $f(x)$ is defined as the following: $$ f(x)= \left\{ \begin{array}{ll} x+2 &, \text{if } x \le -1\\ 2x &, \text{if } -1 < x < 2\\ \displaystyle\frac{x^2}{2} &, \text{if } x \ge 2 \end{array} \right. $$ (A) Compute $f(f(f(-\frac{7}{4})))$ (B) If $f(a)=3$, find the value of $a$

2449

Let $f(x)$ be an odd function and $g(x)$ be an even function. If $f(x)+g(x)=\frac{1}{x-1}$, find $f(x)$ and $g(x)$.

2450

If $f\Big(\displaystyle\frac{x+1}{x}\Big)=\displaystyle\frac{x^2+x+1}{x^2}$, find $f(x)$.

2454

Let $G$ be the centroid of $\triangle{ABC}$, $L$ be a straight line. Prove that $$GG'=\frac{AA'+BB'+CC'}{3}$$ where $A'$, $B'$, $C'$ and $G'$ are the feet of perpendicular lines from $A$, $B$, $C$, and $G$ to $L$.

2461

Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $r$ be the triangle's inradius. Show that $$r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(b+a-c)}{a+b+c}}$$

2472

What is the area that is covered by putting a $8\times 6$ rectangle and a $5 \times 5$ square as shown on a table?

2474

Restaurant MAS offers a set menu with $3$ choices of appetizers, $5$ choices of main dishes, and $2$ choices of desserts. How many possible combinations can a customer have for one appetizer, one main dish, and one dessert?

2475

Eight chairs are arranged in two equal rows for a group of $8$. Joe and Mary must sit in the front row. Jack must sit in the back row. How many different seating plans can they have?

2476

Two Britons, three Americans, and six Chinese form a line:

How many different ways can the $11$ individuals line up?
If two people of the same nationality cannot stand next to each other, how many different ways can the $11$ individuals line up?