ArtOfThinking

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1   
Let $n$ be a positive integer, prove that in this series $$\Big\lfloor{\frac{n}{1}}\Big\rfloor, \Big\lfloor{\frac{n}{2}}\Big\rfloor, \Big\lfloor{\frac{n}{3}}\Big\rfloor \cdots \Big\lfloor{\frac{n}{n}}\Big\rfloor$$, there are less than $2\sqrt{n}$ integers distinct.

50   
Steve is piling $m \geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l \leq n$, such that $i < j$ and $k < l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively,j or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively. Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?

66   
For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

73   

Steve says to Jon, 'I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $$P(x) = 2x^3-2ax^2+(a^2-81)x-c$$

for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?' After some calculations, Jon says, 'There is more than one such polynomial.' Steve says, 'You're right. Here is the value of $a$.' He writes down a positive integer and asks, 'Can you tell me the value of $c$?' Jon says, 'There are still two possible values of $c$.' Find the sum of the two possible values of $c$.


177   
Find that largest integer $A$ that satisfies the following property: in any permutation of the sequence $1001$, $1002$, $1003$, $\cdots$, $2000$, it is always possible to find $10$ consecutive terms whose sum is no less than $A$.

309   
Prove: randomly select 51 numbers from $1, 2, 3, \cdots, 100$, at least two of them must be relatively prime to each other.

314   
There are $n$ points, $A_1$, $A_2$, $\cdots$, $A_n$ on a line segment, $\overline{A_0A_{n+1}}$. The point $A_0$ is black, $A_{n+1}$ is white, and the rest points are colored randomly either black or white. Prove: among these $n+1$ line segments $A_kA_{k+1}$, where $k=0, 1, \cdots, n$, the number of those with different colored ending points is odd.

330   
Joe wrote $3$ positive integers on the whiteboard, then erased one number and replaced with the sum of the other two numbers minus $1$. He continued doing this and stopped when there're $17$, $1967$ and $1983$ on the whiteboard. Is it possible that the initial three numbers Joe wrote are $2$, $2$ and $2$?

341   
Let $a_1$, $a_2$, $a_3$, $\cdots$, $a_n$ be a random permutation of $1$, $2$, $3$, .., $n$, where $n$ is an odd number. Prove $$(a_1-1)(a_2-2)\cdots(a_n-n)$$ is an even number.

364   
A league with $12$ teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores $2$ points for every game it wins and $1$ point for every game it draws. Which of the following is NOT a true statement about the list of $12$ scores?

380   
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6^{th}$ place. Who finished in $8^{th}$ place?

390   

This graph shows the water temperature T degrees at time t minutes for a pot of water placed on a stove and heated to 100 degrees. On average, how many degrees did the temperature increase each minute during the first $8$ minutes?


456   

In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?



459   
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a \geq{1}$ and $a+b+c \leq{7}$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$.

519   
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara's turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins?

533   
When trying to recall some facts about the ages of his three aunts, Josh made the following claims: - Alice is fifteen years younger than twice Catherine's age. - Beatrice is twelve years older than half of Alice's age. - Catherine is eight years younger than Beatrice. - The three women\u2019s ages add to exactly one-hundred years. However, Josh's memory is not perfect, and in fact only three of these four claims are true. If each aunt's age is an integer number of years, how old is Beatrice?

561   
Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?

595   
There are $5$ coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?


615   
Let $R$ be a square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called $n$-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$-ray partitional but not $60$-ray partitional?

629   
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?

649   
A palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $(x+32)$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

656   
For how many integer values of $k$ do the graphs of $x^2+y^2=k^2$ and $xy = k$ not intersect?

672   
A month with $31$ days has the same number of Mondays and Wednesdays.How many of the seven days of the week could be the first day of this month?

754   
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum?


770   
A $4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?


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