Practice (TheColoringMethod)
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Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
Tina randomly selects two distinct numbers from the set $\{ 1, 2, 3, 4, 5 \}$, and Sergio randomly selects a number from the set $\{ 1, 2, ..., 10 \}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$?
The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
Consider the sequence of numbers: $4,7,1,8,9,7,6,\dots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. The smallest value of $n$ for which $S_n>10,000$ is:
Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. What is the probability that $BD > 5\sqrt2$?
Show that among any four randomly selected integers, there must exist two whose difference is a multiple of 3.
Show that it is possible to find an integer whose digits are all 4 and it is a multiple of 2016.
There are 13 randomly selected points inside a square of side length 2. Show that there must exist quadrilateral whose vertice are among these 13 points and area is no more than 1.
Joe randomly selects 50 different numbers from 1, 2, $\cdots$, 97, 98, and finds there are always two of them whose difference is a multiple of 7. Can you explain why?
Among nine randomly selected even numbers from $2$, $4$, $6$, $\cdots$, $28$, $30$, show that at least two of them whose sum is $34$.
Show that, among randomly selected $11$ numbers from $1$, $2$, $3$, $\cdots$, $19$, $20$, one of them must be a multiple of another.
Show that among any 5 integers, three of them must satisfy the condition that their sum is a multiple of 3.
Show that in a $n$-people party, at least two of them have met the same number of other guests before.
There are only two problems in a math test. Ten points will be awarded for every correct answer. Two point will be given for any skipped problem. No point will be given for wrong answer. The teacher claims regardlessly there must be at least 3 students will end up with the same score. Can you figure out the minimal number of students who will take this test?
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?
Two players, $A$ and $B$, take turns naming positive integers, with $A$ playing first. No player may name an integer that can be expressed as a linear combination, with positive integer coefficients, of previously named integers. The player who names 1 loses. Show that no matter how A and B play, the game will always end.
$\textbf{Cards Game (Sum Fifteen)}$
There are nine cards laid out on a table, numbered $1$ through $9$. Two players, Joe and John, take turns to pick up one card a time (and once a card is picked up, it is out of play). As soon as one of these two players has the sum of three of cards equal $15$ , that player wins. Who will win if both players adopt the best strategy?
Alan and Barbara play a game in which they take turns filling entries of an initially empty $1024$ by $1024$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
$\textbf{Heaps of Beans}$
A game starts with four heaps of beans, containing $3$, $4$, $5$ and $6$ beans, respectively. The two players move alternately. A move consists of taking either one bean from a heap, provided at least two beans are left behind in that heap, or a complete heap of two or three beans. The player who takes the last bean wins. Does the first or second player have a winning strategy?
Find the smallest positive integer $n$ so that $107n$ has the same last two digits as $n$.