Practice With Solutions

2679

A total of $2018$ tickets, numbered $1$, $2$, $3$, $\cdots$, $2014$, $2015$ are placed in an empty bag. Alfrid removes ticket $a$ from the bag. Bernice then removes ticket $b$ from the bag. Finally, Charlie removes ticket $c$ from the bag. They notice that $a < b < c$ and $a + b + c = 2018$. In how many ways could this happen?

2681

Show that $$\binom{n}{k} = \frac{n}{k}\binom{n-1}{k-1}$$

2682

Let positive integers $m\le k \le n$. Show that $$\binom{n}{k}\binom{k}{m} =\binom{n}{m}\binom{n-m}{k-m} =\binom{n}{k-m}\binom{n-k+m}{m}$$

2685

Show that $$\binom{n}{0}+\frac{1}{2}\binom{n}{1}+\frac{1}{3}\binom{n}{2}+\cdots+\frac{1}{n+1}\binom{n}{n}=\frac{2^{n+1}-1}{n+1}$$

2686

Using at least two approaches to prove $$\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \cdots +n\binom{n}{n} = n\cdot 2^{n-1}$$

2687

Compute the value of $$\displaystyle\sum_{k=1}^n k^2\binom{n}{k}$$

2688

Simplify: $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2015 \times 2016$

2690

Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$.

2692

Let $X$ be the integer part of $\left(3+\sqrt{7}\right)^n$ where $n$ is a positive integer. Show that $X$ must be odd.

2693

Let $n$ be a positive integer. Show that the smallest integer that is larger than $(1+\sqrt{3})^{2n}$ is divisible by $2^{n+1}$.

2694

Let $m=4k+1$ where $k$ is a non-negative integer. Show that $$a=\binom{n}{1}+m\binom{n}{3}+m^2\binom{n}{5}+\cdots+m^{\frac{n-1}{2}}\binom{n}{n}$$

is divisible by $2^{n-1}$, where $n$ is an odd number.

2696

Show that the following inequality holds for any positive integer $n$: $$(2n+1)^n \ge (2n)^n + (2n-1)^n$$

2697

Let $a$ and $b$ be two positive real numbers. Show that if $\frac{1}{a}+\frac{1}{b}=1$. Prove that the following inequality holds for any positive integer $n$: $$(a+b)^n-a^n-b^n\ge 2^{2n}-2^{n+1}$$

2698

Let $\{a_n\}$ be a sequence defined as $a_n=\lfloor{n\sqrt{2}}\rfloor$ where $\lfloor{x}\rfloor$ indicates the largest integer not exceeding $x$. Show that this sequence has infinitely many square numbers.

2699

Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.

2700

Let the integer and decimal part of $(5\sqrt{2}+7)^{2n+1}$ be $I$ and $D$ respectively. Show that $(I+D)D$ is a constant.

2701

Let $n$ be a non-negative integer. Show that $2^{n+1}$ divides the value of $\left\lfloor{(1+\sqrt{3})^{2n+1}}\right\rfloor$ where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding the give real number $x$.

2702

Let $a$, $b$ be two positive real numbers, and $n$ be a positive integer greater than $2$. Show that $$\frac{a^n+a^{n-1}b+\cdots+ab^{-1}+b^n}{n+1}\ge \Big(\frac{a+b}{2}\Big)^n$$

2703

Show that all the terms of the sequence $a_n=\frac{(2+\sqrt{3})^n-(2-\sqrt{3})^n}{2\sqrt{3}}$ are integers, and also find all the $n$ such that $3 \mid a_n$.

2704

Show that all terms of the sequence $a_n=\left(\frac{3+\sqrt{5}}{2}\right)^n+\left(\frac{3-\sqrt{5}}{2}\right)^n -2$ are integers. And when $n$ is even, $a_n$ can be expressed as $5m^2$, when $n$ is odd $a_n$ can be expressed as $m^2$.

2705

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.

2707

In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?

2708

A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?

2709

Label the first row of the Pascal triangle as row $0$. How many odd numbers are there in the $2019^{th}$ row?

2710

If the sum of all coefficients in the expanded form of $(3x+1)^n$ is $256$, find the coefficient of $x^2$.

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