Evaluate $\displaystyle\lim_{n\to\infty}S_n$ where

$$S_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots + (-1)^{n-1}\frac{1}{n}$$

Determine the values of $\alpha$ and $\beta$ such that 

$$\lim_{n\to\infty}\frac{n^{\alpha}}{n^{\beta}-(n-1)^{\beta}}=2020$$

Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$

For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$

Show that $1-\cos{x} < x^2$ holds for all $x > 0$.

Compute $$\int_0^{\pi}\frac{2x\sin{x}}{3+\cos^2x}dx$$

Compute $$\lim_{x\to\infty}\left[x-x^2\ln\left(\frac{1+x}{x}\right)\right]$$

For a given $x > 0$, let $a_n$ be the sequence defined by $a_1=x$ for $n = 1$ and $a_n = x^{a_{n−1}}$ for $n\ge 2$. Find the largest $x$ for which the limit $\displaystyle\lim_{n\to\infty} a_n$ converges.

Let a differentiable function $f(x)$ satisfy $$f(x)\cos{x} + 2\int_0^xf(t)\sin{t}dt = x+1$$

Find $f(x)$.

Find the value of $\displaystyle\lim_{n\to\infty}\sin^2\left(\pi\sqrt{n^2+n}\right)$.

Find the value of $$I=\int\frac{e^{-\sin{x}}\sin(2x)}{(1-\sin{x})^2}dx$$

Compute $$\int\sqrt{x^2+a^2}dx$$

Evaluate $$\int\frac{5x+6}{(x^2+x+2)^2}dx$$

$\textbf{Toggler's Problem}$

Among a group of $100$ people, only one is a truth teller and the rest $99$ are togglers. A truth teller always tells the truth. A toggler will tell the truth and a lie in an alternating fashion. That is, after he or she tells the truth the first time, this person will tell a lie next time. However, if his or her first answer is false, then the next answer will be true. It is unknown whether a toggler's first answer is the truth or a lie.

If all these people know who is the truth teller, how many questions do you need to ask in order to identify the truth teller?

$\textbf{Color the Grid}$

Two geniuses are playing a game of coloring a $2\times n$ grid where $n$ is an odd integer. Each of them in turn picks a uncolored cell and colors it in either green or red until all the cells are filled. At the end of the game, if the number of adjacent pairs with the same color is greater than the number of adjacent pairs with different colors, then the person who picks and colors first wins the game. (An adjacent pair consists of two cells next to each other.) Otherwise, if there are more adjacent pairs with different colors than those with same color, the person who starts later wins. If these two numbers are the same, the result is a tie. Who will win if both players make no mistake?

$\textbf{Split the Coins}$

There are $100$ regular coins lying flat on a table. Among these coins, $10$ are heads up and $90$ are tails up. You are blindfolded and can not feel, see or in any other way to find out which $10$ are heads up. Is it possible to split the coins into two piles so there are equal numbers of heads-up coins in each pile?

$\textbf{Shatter the Ball}$

You are in a $100$-story building with two identical bowling balls. You want to find the lowest floor at which the ball will shatter when dropped to the ground. What is the minimum number of drops you need in order to find the answer?

$\textbf{Pirates and Gold}$

Five pirates are trying to split up $1000$ gold pieces according to the following rules

• The $1^{st}$ pirate can propose a plan. If his proposal wins the majority's support (including himself), then it is done. Otherwise, he will be instantly killed.
• If the $1^{st}$ pirate is killed, then the $2^{nd}$ pirate will make a proposal. If his proposal wins the majority's support (a tie does not suffice, i.e. he needs at least $3$ votes), it is done. Otherwise, he will be instantly killed.
• The process continues until a proposal is agreed by a majority.

Assuming all these five pirates are intelligent (i.e. always choose the optimal strategy for himself), greedy (i.e. get as much as gold for himself) and ruthless (i.e. the more pirates dead, the better), what will be the final distribution of the gold?

$\textbf{Pirates and Gold (II)}$

What will be the result if all are the same as <myProblem>GetLink/4654</myProblem> except that a proposal only requires $50\%$ of the vote to pass?

$\textbf{Birthday}$

John and Mary know that their boss Joe's birthday is one of the following $10$ dates: Mar $4$, Mar $5$, Mar $8$, Jun $4$, Jun $7$, Sep $1$, Sep $5$, Dec $1$, Dec $2$, and Dec $8$.

Joe tells John only the month of his birthday, and tells Mary only the day.

After that, John first says: “I do not know Joe’s birthday, Mary doesn’t know it either.”

After having heard what John said, Mary says "I did not know Joe's birthday, but I know it now."

After having heard what Mary said, John says "I know Joe's birthday now as well."

What is Joe's birthday?

$\textbf{Casino Game's Fair Price}$

A casino offers a card game using a regular deck of $52$ cards. The rule is that you turn over two cards each time. For each pair, if both cards are black, they go to the dealer’s pile; if both cards are red, they go to your pile; if one card is black and the other is red, they are discarded. The process is repeated until you two go through all $52$ cards. If you have more cards in your pile, you win $\$100$; otherwise (including ties) you get nothing. The casino allows you to negotiate the price you want to pay for the game. How much are you willing to pay to play this game?

$\textbf{Who Paid for the Beer}$

Joe lives in a town along the US-Canada border. One day, both countries' currencies are discounted $10\%$ on the other side of the border. That is, a US dollar is worth $90$ Canadian cents in Canada, and a Canadian dollar is worth $90$ US cents in the US. Joe buys one US dollar's worth of beer in the US. He pays using a ten US dollar bill and receives ten Canadian dollars as exchange. Then, he walks across the border and buys one Canadian dollar's worth of beer. He pays using the ten Canadian dollars he has and receives ten US dollars as exchange. He then walks back to the US side to buy another US dollar's worth of beer. He receives ten Canadian dollars as the exchange before goes to Canada again. After coming back and forth, Joe finally returns to his home and becomes dead drunk. However, he still has ten US dollars in his hand. The question is who has paid for all the beers Joe has consumed?

$\textbf{Secured Delivery}$

John wants to send a valuable gift to Mary. He has a lockable box that is large enough to contain the gift. The box also has a locking ring that can have a few locks attached. However, Mary does not have the key to any of John's locks. How can John send the gift to Mary securely?

Joe and Mary flip a coin ($n+1$) and $n$ times, respectively. What is the probability that Joe gets more heads than Mary does?

$\textbf{Cheating Husbands}$

A remote town comprises of $100$ married couples. Everyone in the town lives by the following rule: If a husband cheats on his wife, the husband is executed at the night as soon as his wife finds out about it. All the women in the town only gossip about husbands of other women. No woman ever tells another woman if that woman's husband is cheating on her. So every woman in the town knows about all the cheating husbands in the town except her own. It can also be assumed that a husband remains silent about his infidelity. One day, the mayor of the town announces to the whole town that there is at least $1$ cheating husband in the town. What will happen afterwards?