Practice (TheColoringMethod)

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A circle of radius $2$, center on the origin, is drawn on a grid of points with integer coordinates. Let $n$ be the grid points that lie within or on the circle. What is the smallest amount of radius needs to increase by for there to be $(2n-5)$ grid points within or on the circle?


A particle moves in the $xy$-plane, starting at the origin $(0, 0)$. At each turn, the particle may move in one of the two ways:

  • it may move two to the right and one up
  • it may move one to the right and two up

What is the closet distance the particle may come to the point $(25, 75)$?


Find the value of $c$ such that two parabolas $y=x^2+c$ and $y^2=x$ touch at a single point.


Explain why we cannot apply the cut-the-rope technique to count the non-negative integer solutions to the equation $$x_1 + x_2 + \cdots + x_k = n$$

For example, can we allow two cuts in the same interval thus to model one of the $x_i$ is zero?


Let $n \ge k$ are two positive integers. Given function $x_1+x_2+\cdots + x_k =n$,

  1. Find the number of positive integer solutions to this equation.
  2. Find the number of non-negative integer solutions to this equation.
  3. Explain the relation between these two cases. i.e. is it possible to derive (2) from (1), and vice versa?

Explain why the count of positive / non-negative integer solutions to the equation $x_1 + x_2 + \cdots + x_k=n$ is equivalent to the case of putting $n$ indistinguishable balls into $k$ distinguishable boxes.


Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?


How many even $4$- digit integers are there whose digits are distinct?


Derive the permutation formula $P_n^n=n\times (n-1)\times\cdots\times 2\times 1$ using the recursion method.


Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} + \sqrt{x} - 2x - 1=0$$


Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} + x - 1=0$$


Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} - x + 1=0$$


Prove that, if $|\alpha| < 2\sqrt{2}$, then there is no value of $x$ for which $$x^2-\alpha|x| + 2 < 0\qquad\qquad(*)$$

Find the solution set of (*)  for $\alpha=3$.

For $\alpha > 2\sqrt{2}$, then the sum of the lengths of the intervals in which $x$ satisfies (*) is denoted by $S$. Find $S$ in terns of $\alpha$ and deduce that $S < 2\alpha$.


Which number is larger: $5^{4321}$ or $4^{5321}$?


Find the minimal value of $4^m + 4^n$ if $m+n=3$.


Show that $2013^2 +2013^2\times 2014^2 + 2014^2$ is a perfect sqare.


Compute $\sqrt[32]{259\times 23\times 11 +9}$.


Let $f(x)$ be a second degree function satisfying $f(-2)=0$ and $2x \lt f(x) \le\frac{x^2+4}{2}$. Find the value of $f(10)$.


Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive numbers. Show that $$\sqrt{a^2+x^2} + \sqrt{b^2 + y^2}+\sqrt{c^2+z^2} \ge \sqrt{(a+b+c)^2 + (x+y+z)^2}$$


Let positive numbers $a$, $b$ and $c$ satisfy $a+b+c=8$. Find the minimal value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$.


Solve $x^{x^{88}} - 88=0$.


Given a pointer pointing to the header of a linked list, how to detect whether the linked list has a loop?

An additional question: how can math knowledge help here?


Show that there exists a perfect sqaure whose leading $2024$ digits are all $1$.


Show that the $(k+1)$ leading digits of the number $\underbrace{333\cdots 3}_{k}4^2$ are all $1$s. Here $k$ is any positive integer.


For any given integer $N$, show that there exists a perfect square whose leading digits match $N$. For example, if $N=2024$, there must exist a perfect square whose leading four digits are $2024$.