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:
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 $x_1+x_2+\cdots + x_k =n$,
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 $2024$.