Prove: there exists a rational number $\frac{c}{d}$, where $d<1000$, such that $$\Big[k\cdot\frac{c}{d}\Big]=\Big[k\cdot\frac{73}{100}\Big]$$ holds for every positive integer $k$ that is less than 1000. Here $\Big[x\Big]$ denotes the largest integer that is not exceeding $x$.
Seven students count from $1$ to $1000$ as follows:
- Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1$, $3$, $4$, $6$, $7$, $9$, . . ., $997$, $999$, $1000$.
- Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
- Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
- Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
- Finally, George says the only number that no one else says.
What number does George say?
The number obtained from the last two non-zero digits of $90!$ is equal to $n$. What is $n$?
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^{\circ}$, $\angle CDE = 168^{\circ}$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$.
$\textbf{Lying Politicians}$
Suppose $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside him or her are both liars. What are the maximum possible number and minimum possible number of liars?
Let integer $n\ge 2$, prove $$\sin{\frac{\pi}{n}}\cdot\sin{\frac{2\pi}{n}}\cdots\sin{\frac{(n-1)\pi}{n}}=\frac{n}{2^{n-1}}$$
Let $A=x\cos^2{\theta} + y\sin^2{\theta}$, $B=x\sin^2{\theta}+y\sin^2{\theta}$, where $x$, $y$, $A$, and $B$ are all real numbers. Prove $x^2 + y^2 \ge A^2 + B^2$
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{(2k-1)^2+a_k^2}$, where $n\in\mathbb{N}$, $a_1, a_2, \cdots, a_n\in\mathbb{R}^+$, and $a_1+a_2+\cdots a_n = 17$. If there exists a unique $n$ such that $S_n$ is also an integer, find $n$.
Solve in integers $y^2=x^4 + x^3 + x^2 +x +1$.
Solve in integers $x^3 + (x+1)^3 + \cdots + (x+7)^3 = y ^3$
$N$ delegates attend a round-table meeting, where $N$ is an even number. After a break, these delegates randomly pick a seat to sit down again to continue the meeting. Prove that there must exist two delegates so that the number of people sitting between them is the same before and after the break.
Prove there is no integer solutions to $x^2 = y^5 - 4$.
Let $\alpha$ and $\beta$ be two real roots of the equation $x^2 + x - 4=0$. Find the value of $\alpha^2 - 5\beta + 10$ without computing the value of $\alpha$ and $\beta$.
Find all triangles whose sides are consecutive integers and areas are also integers.
Find all positive integers $k$, $m$ such that $k < m$ and
$$1+ 2 +\cdots+ k = (k +1) + (k + 2) +\cdots+ m$$
How many different ways to express $13$ as the sum of several positive odd numbers? Order
matters. For example, $1 + 1 + 3 + 3 + 5$ is treated as a different expression as 1 + 3 + 1 + 3 + 5
For a positive integer m, we define $m$ as a $\textit{factorial}$ number if and only if there exists a positive integer $k$ for which $m = k\cdot(k - 1)\cdots 2\cdot 1$. We define a positive integer $n$ as a $\textit{Thai}$ number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers.
What is the sum of the five smallest $\textit{Thai}$ numbers?
If $a_0, a_1,\cdots, a_n \in \{0, 1, 2,\cdots, 9\}, n\ge 1, a_0\ge 1$, then the zeros of $f(x)=a_0 x^n + a_1x^{n-1} +\cdots +a_n$ have real parts less then 4.
Let complex numbers $a$, $b$, and $c$ satisfy $a|bc| + b|ca| + c|ab| = 0$. Show that $$|(a-b)(b-c)(c-a)|\ge 3\sqrt{3}|abc|$$
Let $x, y \in \big(0, \frac{\pi}{2}\big)$. Show that if the equation $(\cos x + i \sin y)^n = \cos nx + i \sin ny$ holds for two consecutive positive integers, then it will hold for all positive integers.
Find all polynomials $f(x)$ such that $f(x^2) = f(x)f(x+1)$.
Let $\gamma_i$ and $\overline{\gamma_i}$ be the 10 zeros of $x^{10}+(13x-1)^{10}$, where $i=1, 2, 3, 4, 5$. Compute $$\frac{1}{\gamma_1 \overline{\gamma_1}}+\frac{1}{\gamma_2 \overline{\gamma_2}}+\cdots+\frac{1}{\gamma_5 \overline{\gamma_5}}$$
Show that $$\sin\frac{\pi}{2n+1}\cdot\sin\frac{2\pi}{2n+1}\cdots\sin\frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$$
In baseball, a player's batting average is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $0.399$. What is the fewest number of at bets that Danielle could have?
Prove that if $p$ and $(p^2 + 8)$ are prime, then $(p^3 + 8p + 2)$ is prime.