Find the largest integer less than $2012$ all of whose divisors have at most two $1$’s in their binary representations. In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.

What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?

Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation: Samuel Tsui: Do our numbers have any common factors greater than $1$? Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$. Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal. What is the sumof Samuel Tsui’s and Jason Yang’s numbers? [i]Proposed by Samuel Tsui[/i]

For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $2018 - 3018 + 4018$? [b]p2.[/b] What is the smallest integer greater than $100$ that is a multiple of both $6$ and $8$? [b]p3.[/b] What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express your answer as a decimal. [b]p4.[/b] A non-degenerate triangle has sides of lengths $1$, $2$, and $\sqrt{n}$, where $n$ is a positive integer. How many possible values of $n$ are there? [b]p5.[/b] When three integers are added in pairs, and the results are $20$, $18$, and $x$. If all three integers sum to $31$, what is $x$? [b]p6.[/b] A cube's volume in cubic inches is numerically equal to the sum of the lengths of all its edges, in inches. Find the surface area of the cube, in square inches. [b]p7.[/b] A $12$ hour digital clock currently displays$ 9 : 30$. Ignoring the colon, how many times in the next hour will the clock display a palindrome (a number that reads the same forwards and backwards)? [b]p8.[/b] SeaBay, an online grocery store, offers two different types of egg cartons. Small egg cartons contain $12$ eggs and cost $3$ dollars, and large egg cartons contain $18$ eggs and cost $4$ dollars. What is the maximum number of eggs that Farmer James can buy with $10$ dollars? [b]p9.[/b] What is the sum of the $3$ leftmost digits of $\underbrace{999...9}_{2018\,\,\ 9' \,\,s}\times 12$? [b]p10.[/b] Farmer James trisects the edges of a regular tetrahedron. Then, for each of the four vertices, he slices through the plane containing the three trisection points nearest to the vertex. Thus, Farmer James cuts off four smaller tetrahedra, which he throws away. How many edges does the remaining shape have? [b]p11.[/b] Farmer James is ordering takeout from Kristy's Krispy Chicken. The base cost for the dinner is $\$14.40$, the sales tax is $6.25\%$, and delivery costs $\$3.00$ (applied after tax). How much did Farmer James pay, in dollars? [b]p12.[/b] Quadrilateral $ABCD$ has $ \angle ABC = \angle BCD = \angle BDA = 90^o$. Given that $BC = 12$ and $CD = 9$, what is the area of $ABCD$? [b]p13.[/b] Farmer James has $6$ cards with the numbers $1-6$ written on them. He discards a card and makes a $5$ digit number from the rest. In how many ways can he do this so that the resulting number is divisible by $6$? [b]p14.[/b] Farmer James has a $5 \times 5$ grid of points. What is the smallest number of triangles that he may draw such that each of these $25$ points lies on the boundary of at least one triangle? [b]p15.[/b] How many ways are there to label these $15$ squares from $1$ to $15$ such that squares $1$ and $2$ are adjacent, squares $2$ and $3$ are adjacent, and so on? [img]https://cdn.artofproblemsolving.com/attachments/e/a/06dee288223a16fbc915f8b95c9e4f2e4e1c1f.png[/img] [b]p16.[/b] On Farmer James's farm, there are three henhouses located at $(4, 8)$, $(-8,-4)$, $(8,-8)$. Farmer James wants to place a feeding station within the triangle formed by these three henhouses. However, if the feeding station is too close to any one henhouse, the hens in the other henhouses will complain, so Farmer James decides the feeding station cannot be within 6 units of any of the henhouses. What is the area of the region where he could possibly place the feeding station? [b]p17.[/b] At Eggs-Eater Academy, every student attends at least one of $3$ clubs. $8$ students attend frying club, $12$ students attend scrambling club, and $20$ students attend poaching club. Additionally, $10$ students attend at least two clubs, and $3$ students attend all three clubs. How many students are there in total at Eggs-Eater Academy? [b]p18.[/b] Let $x, y, z$ be real numbers such that $8^x = 9$, $27^y = 25$, and $125^z = 128$. What is the value of $xyz$? [b]p19.[/b] Let $p$ be a prime number and $x, y$ be positive integers. Given that $9xy = p(p + 3x + 6y)$, find the maximum possible value of $p^2 + x^2 + y^2$. [b]p20.[/b] Farmer James's hens like to drop eggs. Hen Hao drops $6$ eggs uniformly at random in a unit square. Farmer James then draws the smallest possible rectangle (by area), with sides parallel to the sides of the square, that contain all $6$ eggs. What is the probability that at least one of the $6$ eggs is a vertex of this rectangle? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$. [i]S. Berlov[/i]

[b]a.)[/b] Let $m>1$ be a positive integer. Prove there exist finite number of positive integers $n$ such that $m+n|mn+1$. [b]b.)[/b] For positive integers $m,n>2$, prove that there exists a sequence $a_0,a_1,\cdots,a_k$ from positive integers greater than $2$ that $a_0=m$, $a_k=n$ and $a_i+a_{i+1}|a_ia_{i+1}+1$ for $i=0,1,\cdots,k-1$.

$A$ is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of $A$ also belongs to $A$.

Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\] [i]Proposed by Holden Mui[/i]

For a positive integer $n$, prove that $$\sum_{n \le p \le n^4} \frac{1}{p} < 4$$ where the sum is taken across primes $p$ in the range $[n, n^4]$ [i]N. Filonov[/i]

Find all pairs of integers $(m, n)$ which satisfy the equation \[(2n^2+5m-5n-mn)^2=m^3n\]

$a_n$ sequence is a arithmetic sequence with all terms be positive integers. (for $a_n$ non-constant sequence) Let $p_n$ is greatest prime divisor of $a_n$. Prove that $$(\frac{a_n}{p_n})$$ sequence is infinity. [hide]Note: If we find a $M>0$ constant such that $x_n \leq M$ for all $n \in {\mathbb N}$'s, $(x_n)$ sequence is non-infinite, but we can't find $M$, $(x_n)$ sequence is infinity [/hide]

All the positive divisors of a positive integer $n$ are stored into an increasing array. Mary is writing a programme which decides for an arbitrarily chosen divisor $d > 1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\left\lceil\frac{k}{2}\right\rceil$ divisors of $n$: $d$ is prime if and only if none of them but $1$ divides $d$. Is Mary right?

The nine digits from 1 to 9 are to be placed around a circle so that the average of any three consecutive digits is a multiple of 3. Is this possible? Justify your answer.

Prove that there exists a number divisible by $5^{1000}$ not containing a single zero in its decimal notation.

Let $n$ be a positive integer. On a board there are written all integers from $1$ to $n$. Alina does $n$ moves consecutively: for every integer $m$ $(1 \leq m \leq n)$ the move $m$ consists in changing the sign of every number divisible by $m$. At the end Alina sums the numbers. Find this sum.

Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.

If we divide number $1^{1990}+2^{1990}+3^{1990}+...+1990^{1990}$ with $10$, what remainder will we find?

Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$.

Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)

Find the smallest positive integer $N$ of two or more digits that has the following property: If we insert any non-null digit $d$ between any two adjacent digits of $N$ we obtain a number that is a multiple of $d$.

If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$