Practice (137)

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There are $7$ boys each of which has at least $3$ brothers among the other $6$ boys. Are these $7$ boys necessarily all brothers? Explain.

$\textbf{Lily Pads}$

There are $24$ lily pads shown below. A toad can jump from one pad to an adjacent one either horizontally or vertically, but not diagonally. Can this toad visit all the pads without stopping at a pad for more than once? It can choose any pad to start its journey.


Find the largest multiple of 99 among the nine-digit integers, whose digits are all distinct.

What is the maximum number of acute triangles 2n + 1 lines can create?

There are four masses all whose weights are all integers. The total weight of these masses is 40$g$. If it is possible to measure any integer weight between 1$g$ and 40$g$ using some combinations of these masses, what are their weights respectively?

How many distinct positive integers can be expressed in the form $ABCD-DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? (Here $A, B, C$ and $D$ are digits, possible equal)

Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the value of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgian, one card to each, and asked them to figure out which trigonometric function (sin, cos, tan) produced their cards. Even after sharing the values on their cards with each other, only Malvian was able to surly identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Marlvina's card.

Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$.

Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB each, 12 of the files take up 0.7 MB each, and the rest take up 0.4 MB each. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?

A $3\times 3\times 3$ cube is made of 27 normal dice. Each die's opposite sides sum to 7. What is the smallest possible sum of all of the values visible on the 6 faces of the large cube?

There are four wolves standing on the four corners of a square, and a rabbit standing at the center of that square. If a wolf can run at $1.4$ times of the rabbit's speed, but can only move along the sides of this square, can the rabbit escape to outside the square?

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game?

What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?

Tom and Jerry are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2,\cdots, 2014$ are to be written. (Each number will be written exactly once). Tom fills in a piece of paper first. How many pieces of paper must Jerry fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Tom's in at least one position?

Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when takes a blue pill, he will loose one pound. If Neo originally wights one pound, what is the minimum number of pills he must take to make his weight 2015 pound?

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).

Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$, then the slip with 3.5 must go into what cup?

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?


Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.)

  • Find, with proof, the maximum possible value of $n$.
  • For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard.

As shown in the figure, in a regular triangular house, all the rooms are in the shape of regular triangles. There are doors between adjacent rooms. Starting from one of the rooms, go through the doors to visit other rooms, without repeating rooms or leaving the house. Including the starting room, how many rooms can be visited?


Mr Wise come across a group of $4$ people. He finds that some of these $4$ people always tell the truth and some always tell lies. So he asks each of them and gets the following answers:

  • $A$: "All of us tell lies."
  • $B$: "There is only one among our four who tells lies."
  • $C$: "There are two among us who tell lies."
  • $D$: "I am telling the truth."

Do you think whether $D$ tells the truth?


Show that if an $m\times n$ grid can be completely covered by some L-shaped smaller grids consist of 4 unit grids without overlapping, then the value of $mn$ must be a multiple of 8.

In chess, a knight can move between the two opposite corner squares of a $2 \times 3$ block. The $2\times 3$ block can be either horizontal or vertical, and can be either direction of where the knight stands. In a $4\times 4$ grid, how many squares can a knight visit, including its starting square, in a series of moves without stopping at the same square twice?

Show that it is impossible to cover an $8\times 8$ square using fifteen $4\times 1$ rectangles and one $2\times 2$ square.

Is it possible to arrange these numbers, $1, 1, 2, 2, 3, 3, \cdots, 1986, 1986$ to form a sequence for such there is $1$ number between two $1$'s, $2$ numbers between two $2$'s, $\cdots$, $1986$ numbers between two 1986's?