If the $5^{th}$, $6^{th}$ and $7^{th}$ coefficients in the expansion of $(x^{-\frac{4}{3}}+x)^n$ form an arithmetic sequence, find the constant term in the expanded form.
If the sum of all coefficients in the expanded form of $(3x+1)^n$ is $256$, find the coefficient of $x^2$.
Find the coefficient of the $x$ term after having expanded $$(x^2+3x+2)^5$$
Find the constant term after $\left(\mid x\mid +\frac{1}{\mid x \mid} -2\right)^2$ is expanded.
What is the remainder when $2021^{2020}$ is divided by $10^4$?
Show that $$1+4\binom{n}{1} + 7\binom{n}{2}+\cdots+(3n+1)\binom{n}{n}=(3n+2)\cdot 2^{n-1}$$
How many fractions in simplest form are there between $0$ and $1$ such that the products of their denominators and numerators equal $20!$?
A positive integer is written on each face of a cube. Then for each vertex of the cube, the product of the numbers on the three faces associated with this vertex is calculated. If the sum of these eight products equals 2015, find the sum of all the numbers on the 6 faces.
How many fraction numbers between $0$ and $1$ are there whose denominator is $1001$ when written in its simplest form?
How many positive integers not exceeding $10^6$ are there which are neither square nor cubic?
There are $7$ pieces of paper on a table. Cut some of them into $7$ each and put them back to the table. Repeat this process as many times as you wish. Is it possible to have $1990$ pieces of paper on the table at some moment?
The germination rates of two different seeds are measured at $90\%$ and $80\%$, respectively. Find the probability that
- both will germinate
- at least one will germinate
- exactly one will germinate
A bug crawls from $A$ along a grid. It never goes backward, it crawls towards all the other possible directions with equal probability. For example:
- At $A$, it may crawl to either $B$ or $D$ with a 50-50 chance
- At $E$ (coming from $D$), it may crawl to $B$, $F$, or $H$ with a $\frac{1}{3}$ chance each
- At $C$ (coming from $B$), it will crawl to $F$ for sure
The questions are, from $A$:
- What is the probability of it landing at $E$ in 2 steps?
- What is the probability of it landing at $F$ in 3 steps?
- What is the probability of it landing at $G$ in 4 steps?
The probability that Alice can solve a given problem is $1/2$. Beth has $1/3$ chance to solve the same problem. Carol's chance to solve it is $1/4$. If all them work on this problem independently, what is the probability that one and only one of them solves it?
What is the last digit of $7^{222}$?
There are $6$ points in the $3$-D space. No three points are on the same line and no four points are one the same plane. Hence totally $15$ segments can be created among these points. Show that if each of these $15$ segments is colored either black or white, there must exist a triangle whose sides are of same color.
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
Among nine randomly selected even numbers from $2$, $4$, $6$, $\cdots$, $28$, $30$, show that at least two of them whose sum is $34$.
A chocolate bar is made up of a rectangular $m\times n$ grid of small squares. Two players take turns breaking up the bar. On a given turn, a player picks a rectangular piece of chocolate and breaks it into two smaller rectangular pieces, by snapping along one whole line of subdivisions between its squares. The player who makes the last break wins. Does one of the players have a winning strategy for this game?
Let $A_n$ be the average of all the integers between 1 and 101 which are the multiples of $n$ . Which is the largest among $A_2, A_3, A_4, A_5$ and $A_6$?
$\textbf{Passing the Bridge}$
It is a dark and stormy night. Four people must evacuate from an island to the mainland. The only link is a narrow bridge which allows passage of two people at a time. Moreover, the bridge must be illuminated, and the four people have only one lantern among them. After each passage to the mainland, if there are still people on the island, someone must bring the lantern back. When they cross the bridge individually, the four people take $2$, $4$, $8$ and $16$ minutes, respectively. Crossing the bridge in pairs, the slower speed is used. What is the minimum time for the entire evacuation?
In triangle $ABC$, $E$ is a point on $AC$ and $F$ is a point on $AB$. $BE$ and $CF$ intersect at $D$. If the areas of triangles $BDF$, $BCD$ and $CDE$ are 3, 7 and 7 respectively, what is the area of the quadrilateral $AEDF$?
The sum of 2008 consecutive positive integers is a perfect square. What is the minimum value of the largest of these integers?
Find the largest positive integer $n$ such that $(3^{1024} - 1)$ is divisible by $2^n$.
A farmer has four straight fences, with respective lengths 1, 4, 7 and 8 metres. What is the maximum area of the quadrilateral the farmer can enclose?