Practice (Intermediate)
back to index | new
There are $2$ white balls, $3$ red balls, and $1$ yellow balls in a jar. How many different ways are there to retrieve $3$ balls to form a line?
What is the last digit of $17^{17^{17^{17}}}$?
Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\cdots$, $8$, $1$, $2$, $3$, $\cdots$, $8$, $\cdots$ in order. Line segments may only be drawn to connect points labeled with the same number. What is the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).
Consider an orange and black coloring of a $20\times 14$ square grid. Let $n$ be the number of coloring such that every row and column has an even number of orange square. Evaluate $\log_2 n$.
Find the number of fractions in the following list that is in its lowest form (i.e. the denominator and the numerator are co-prime). $$\frac{1}{2014}, \frac{2}{2013}, \frac{3}{2012}, \cdots, \frac{1007}{1008}$$
For all positive integer $n$, show that $$\sum_{k=1}^n\frac{k\cdot k! \cdot\binom{n}{k}}{n^k}=n$$
To build roads between $16$ cities so that one can travel from any city to any other city by passing through at most one other city. What is the minimum number of roads that the city with the most road has?
Let $m$ and $n$ be two positive integers between $2$ and $99$, inclusive. Mr. $S$ knows their sum, and Mr. $P$ knows their product. Following are their conversations:
- Mr. $S$: I am certain that you don't know these two numbers individually. But I don't know them either.
- Mr. $P$: Yes, I didn't know. But I know them now.
- Mr. $S$: If this is the case, I know them now too.
What are the two numbers?
Factorize: $f(a)=4a^4-3a^3-2a^2+3a-2$
Factorize: $(ab+bc+ca)(a+b+c)-abc+(a+b)(b+c)(c+a)$
Factorize: $f(x,y,z)=(x+y+z)^5-x^5-y^5-z^5$
Factorize $f(x,y,z) = x^3+y^3 +z^3 - 3xyz$.
Simplify $$\frac{(y-z)^3 +(z-x)^3+(x-y)^3}{(y-z)(z-x)(x-y)}$$
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2mx + (m^2+2m+3)=0$. Find the minimal value of $x_1^2 + x_2^2$.
How many integer pairs $(a,b)$ with $1 < a, b\le 2015$ are there such that $log_a b$ is an integer?
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n =\frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
A word is an ordered, non-empty sequence of letters, such as word or wrod. How many distinct words can be made from a subset of the letters $\textit{c, o, m, b, o}$, where each letter in the list is used no more than the number of times it appears?
For every integer $n$, let $m$ denote the integer made up of the last four digit of $n^{2015}$. Consider all positive integer $n < 10000$, let $A$ be the number of cases when $n > m$, and $B$ be the number of cases when $n < m$. Compute $A-B$.
Let $\frac{p}{q}=1+ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100000}$ where $p$ and $p$ are both positive integers and do not have common divisor greater than 1. How many ending zeros does $q$ have?
(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$
Solve $4x^2+27x-9\equiv 0\pmod{15}$
Solve $5x^3 -3x^2 +3x-1\equiv 0\pmod{11}$
Solve $14x\equiv 30 \pmod{21}$
Solve $17x\equiv 229\pmod{1540}$.
Show that the sum of all the numbers of the form $\frac{1}{mn}$ is not an integer, where $m$ and $n$ are integers, and $1\le m \le n \le 2017$.