Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$

Suppose that a rectangle with sides $ a$ and $ b$ is arbitrarily cut into $ n$ squares with sides $ x_{1},\ldots,x_{n}$. Show that $ \frac{x_{i}}{a}\in\mathbb{Q}$ and $ \frac{x_{i}}{b}\in\mathbb{Q}$ for all $ i\in\{1,\cdots, n\}$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

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.

Let $ ABC$ be an acute triangle. Points $ D$, $ E$, and $ F$ lie on segments $ BC$, $ CA$, and $ AB$, respectively, and each of the three segments $ AD$, $ BE$, and $ CF$ contains the circumcenter of $ ABC$. Prove that if any two of the ratios $ \frac{BD}{DC}$, $ \frac{CE}{EA}$, $ \frac{AF}{FB}$, $ \frac{BF}{FA}$, $ \frac{AE}{EC}$, $ \frac{CD}{DB}$ are integers, then triangle $ ABC$ is isosceles.

Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]

$a_{1},a_{2},\dots,a_{100}$ are real numbers such that:\[ a_{1}\geq a_{2}\geq\dots\geq a_{100}\geq0\] \[ a_{1}^{2}+a_{2}^{2}\geq100\] \[ a_{3}^{2}+a_{4}^{2}+\dots+a_{100}^{2}\geq100\] What is the minimum value of sum $a_{1}+a_{2}+\dots+a_{100}.$

The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$

[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].

There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.” After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made. Can the students come up with a strategy that guarantees a “passed” mark to each of them? [i] Denis Afrizonov [/i]

Assume that $C$ is a right angle of triangle $ABC$ and $N$ is a midpoint of the semicircle, constructed on $CB$ as on diameter externally. Prove that $AN$ divides the bisector of angle $C$ in half.

$23$ frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave $\frac{1}{2}$ and $\frac{1}{6}$ of his water to his left and right neighbors respectively on the first turn and kept $\frac{1}{3}$ for himself. On each subsequent turn everyone gives $\frac{1}{2}$ and $\frac{1}{6}$ of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After $23$ turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn? (Note: you may find it useful to know that $1+x+x^2+\cdot +x^{23}$ has no polynomial factors with rational coefficients)

We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

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

Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$

Solve the equation $\sqrt{7-x}=7-x^2$, where $x>0$.

Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that \[\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.\]

4.4. Prove that there exists a set X of 1996 positive integers with the following properties: (i) the elements of X are pairwise coprime; (ii) all elements of X and all sums of two or more distinct elements of X are composite numbers

Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \] [i]Viorel Vajaitu[/i]

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.

Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.