Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\ (1) $f$ has periodic $2017$\\ (2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\ Proposed by William Chao

Prove that there are infinitely many quadruplets of positive integers $(a,b,c,d)$, such that\\ $ab+1$, $bc+16$, $cd+4$, $ad+9$\\ are perfect squares

Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

We say that a positive integer is super odd if all of its digits are odd. For example, 1737 is super odd and 3051 is not. Find an even positive integer that cannot be express as a sum of two super odd numbers and explain why it is not possible to express it thus.

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$? If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.

Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter. $i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation. $ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.

Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers. Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

Determine all triples $(m , n , p)$ satisfying : \[n^{2p}=m^2+n^2+p+1\] where $m$ and $n$ are integers and $p$ is a prime number.

There are $ 2010 $ people sitting around a round table. First, we give one person $ x $ a candy. Next, we give candies to $1$ st person, $1+2$ th person, $ 1+2+3$ th person, $\cdots$ , and $1+2+\cdots + 2009 $ th person clockwise from $ x $. Find the number of people who get at least one candy.

[u]Round 5[/u] [b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon). When she looks through lens $B$, she sees the shape with $3$ fewer sides (so a hexagon would look like a triangle). How many sides are in the shape that has $200$ more diagonals when looked at from lense $A$ than from lense $B$? [b]p14.[/b] How many ways can you choose $2$ cells of a $5$ by $5$ grid such that they aren't in the same row or column? [b]p15.[/b] If $a + \frac{1}{b} = (2015)^{-1}$ and $b + \frac{1}{a} = (2016)^2$ then what are all the possible values of $b$? [u]Round 6[/u] [b]p16.[/b] In Canadian football, linebackers must wear jersey numbers from $30 -35$ while defensive linemen must wear numbers from $33 -38$ (both intervals are inclusive). If a team has $5$ linebackers and $4$ defensive linemen, how many ways can it assign jersey numbers to the $9$ players such that no two people have the same jersey number? [b]p17.[/b] What is the maximum possible area of a right triangle with hypotenuse $8$? [b]p18.[/b] $9$ people are to play touch football. One will be designated the quarterback, while the other eight will be divided into two (indistinct) teams of $4$. How many ways are there for this to be done? [u]Round 7[/u] [b]p19.[/b] Express the decimal $0.3$ in base $7$. [b]p20.[/b] $2015$ people throw their hats in a pile. One at a time, they each take one hat out of the pile so that each has a random hat. What is the expected number of people who get their own hat? [b]p21.[/b] What is the area of the largest possible trapezoid that can be inscribed in a semicircle of radius $4$? [u]Round 8[/u] [b]p22.[/b] What is the base $7$ expression of $1211_3 \cdot 1110_2 \cdot 292_{11} \cdot 20_3$ ? [b]p23.[/b] Let $f(x)$ equal the ratio of the surface area of a sphere of radius $x$ to the volume of that same sphere. Let $g(x)$ be a quadratic polynomial in the form $x^2 + bx + c$ with $g(6) = 0$ and the minimum value of $g(x)$ equal to $c$. Express $g(x)$ as a function of $f(x)$ (e.g. in terms of $f(x)$). [b]p24.[/b] In the country of Tahksess, the income tax code is very complicated. Citizens are taxed $40\%$ on their first $\$20, 000$ and $45\%$ on their next $\$40, 000$ and $50\%$ on their next $\$60, 000$ and so on, with each $5\%$ increase in tax rate aecting $\$20, 000$ more than the previous tax rate. The maximum tax rate, however, is $90\%$. What is the overall tax rate (percentage of money owed) on $1$ million dollars in income? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. .Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$. i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$. ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.

Let be a natural number $ a\ge 2. $ Prove that for any choice of primes which has the property that none of them divides any of the numbers $ N_n=1+a+a^2+a^3+\cdots +a^{2n} , $ with natural $ n, $ there is another prime not among this choice which doesn't divide any of the numbers $ N_n. $

Let $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $0$ through $9$ such that no two adjacent digits are consecutive. Find $M - m$.

A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.

Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$? A. $4$ B. $6$ C. $8$ D. $10$ E. $12$

Find all integer solutions to \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.\]

Find the number of positive integers less than or equal to $2020$ that are relatively prime to $588$.

Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$

There are $13$ stones each of which weighs an integer number of grams. It is known that any $12$ of them can be put on two pans of a balance scale, six on each pan, so that they are in equilibrium (i.e., each pan will carry an equal total weight). Prove that either all stones weigh an even number of grams or all stones weigh an odd number of grams.

Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

When $a_0, a_1, \dots , a_{1000}$ denote digits, can the sum of the $1001$-digit numbers $a_0a_1\cdots a_{1000}$ and $a_{1000}a_{999}\cdots a_0$ have odd digits only?