Practice With Solutions

2441

Find the range of the function $$y=x+\sqrt{x^2 -3x+2}$$

2443

For any non-negative real numbers $x$ and $y$, the function $f(x+y^2)=f(x) + 2[f(y)]^2$ always holds, $f(x)\ge 0$, $f(1)\ne 0$. Find the value of $f(2+\sqrt{3})$.

2444

Let $f(x)$ be a function defined on $\mathbb{R}$. If for every real number $x$, the relationships $$f(x+3)\le f(x)+3\quad\text{and}\quad f(x+2)\ge f(x)+2$$ always hold. 1) Show $g(x) = f(x)-x$ is a periodic function. 2) If $f(998)=1002$, compute $f(2000)$

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 $$r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(b+a-c)}{a+b+c}}$$

2470

Colour all the points on a plane either white or black randomly. Show that it is always possible to find a triangle whose vertices have the same colour and its side length is either $1$ or $\sqrt{3}$.

2471

Randomly colour all the points one a plane either black or white. Show that if any two points with a distance of $2$ units have the same colour, then all the points on this plane have the same colour.

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?

2479

How many different $6$-digit numbers can be formed by using digits $1$, $2$, and $3$, if no adjacent digits can be the same?

2480

Joe wants to write $1$ to $n$ in a $1 \times n$ grid. The number 1 can be written in any grid, while the number $2$ must be written next to $1$ (can be at either side) so that these two numbers are together. The number 3 must be written next to this two-number block. This process goes on. Every new number written must stay next to the existing number block. How many different ways can Joe fill this $1 \times n$ grid?

2481

How many positive divisors does $20$ have?

2482

Find the number of different rectangles that satisfy the following conditions:

Its area is $2015$
The lengths of all its sides are integers

2483

How many integer solutions does the equation $(x+1)(y+1)=25$ have?

2493

$\textbf{Cutting Pizza}$

Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of pieces (the pizza is too heavy to be lifted up). How many finite pieces of pizza do you have?

2495

How many different ways are there to cover a $1\times 10$ grid with some $1\times 1$ and $1\times 2$ pieces without overlapping?

2505

How many numbers between $1$ and $2020$ are multiples of $3$ or $4$ but not $5$?

2506

How many positive integers, not exceeding $2019$, are relatively prime to $2019$?

2507

Let $p$ be a prime number, computer $\varphi(p)$.

2508

Let $p$ be a prime number and $n$ be a positive integer. Show that $\varphi(p^n)=p^n - p^{n-1}$ where $\varphi(n)$ is the Euler's totient function.

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