For a convex polygon $P$ in the plane let $P'$ denote the convex polygon with vertices at the midpoints of the sides of $P.$ Given an integer $n \geq 3,$ determine sharp bounds for the ratio \[ \frac{\text{area } P'}{\text{area } P}, \] over all convex $n$-gons $P.$

Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

The rectangular faces of rectangular prism $A$ have perimeters $12$, $16$, and $24$. The rectangular faces of rectangular prism $B$ have perimeters $12$, $16$, and $20$. Let $V_A$ denote the volume of $A$ and $V_B$ denote the volume of $B$. Find $V_A-V_B$.

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? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

[u]Round 1[/u] [b]p1.[/b] Today, the date $4/9/16$ has the property that it is written with three perfect squares in strictly increasing order. What is the next date with this property? [b]p2.[/b] What is the greatest integer less than $100$ whose digit sumis equal to its greatest prime factor? [b]p3.[/b] In chess, a bishop can only move diagonally any number of squares. Find the number of possible squares a bishop starting in a corner of a $20\times 16$ chessboard can visit in finitely many moves, including the square it stars on. [u]Round 2 [/u] [b]p4.[/b] What is the fifth smallest positive integer with at least $5$ distinct prime divisors? [b]p5.[/b] Let $\tau (n)$ be the number of divisors of a positive integer $n$, including $1$ and $n$. Howmany positive integers $n \le 1000$ are there such that $\tau (n) > 2$ and $\tau (\tau (n)) = 2$? [b]p6.[/b] How many distinct quadratic polynomials $P(x)$ with leading coefficient $1$ exist whose roots are positive integers and whose coefficients sum to $2016$? [u]Round 3[/u] [b]p7.[/b] Find the largest prime factor of $112221$. [b]p8.[/b] Find all ordered pairs of positive integers $(a,b)$ such that $\frac{a^2b^2+1}{ab-1}$ is an integer. [b]p9.[/b] Suppose $f : Z \to Z$ is a function such that $f (2x)= f (1-x)+ f (1-x)$ for all integers $x$. Find the value of $f (2) f (0) +f (1) f (6)$. [u]Round 4[/u] [b]p10.[/b] For any six points in the plane, what is the maximum number of isosceles triangles that have three of the points as vertices? [b]p11.[/b] Find the sum of all positive integers $n$ such that $\sqrt{n+ \sqrt{n -25}}$ is also a positive integer. [b]p12.[/b] Distinct positive real numbers are written at the vertices of a regular $2016$-gon. On each diagonal and edge of the $2016$-gon, the sum of the numbers at its endpoints is written. Find the minimum number of distinct numbers that are now written, including the ones at the vertices. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A right-angled triangle has an angle equal to $30^\circ.$ Prove that one of the bisectors of the triangle is twice as short as another one. [i]Egor Bakaev[/i]

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB\plus{}BC$. Prove that $ \angle ABC>120^\circ$.

A sequence is defined as $a_1=2025$ and for all $n\ge 2$, $$a_n=\frac{a_{n-1}+1}{n}$$ Determine the smallest $k$ such that $\displaystyle a_k<\frac{1}{2025}$. [i]Proposed by Ivan Chan Kai Chin[/i]

In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and \[a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1\] Prove that \[52 < a_{1371} < 65\]

Show that if we choose any $n + 2$ distinct numbers from the set $\{1, 2, 3, . . . , 3n\}$ there will be two whose difference is greater than $n$ and smaller than $2n$.

[b]p1.[/b] An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days? [b]p2.[/b] A square “rolls” inside a circle of area $\pi$ in the obvious way. That is, when the square has one corner on the circumference of the circle, it is rotated clockwise around that corner until a new corner touches the circumference, then it is rotated around that corner, and so on. The square goes all the way around the circle and returns to its starting position after rotating exactly $720^o$. What is the area of the square? [b]p3.[/b] How many ways are there to fill a $3\times 3$ grid with the integers $1$ through $9$ such that every row is increasing left-to-right and every column is increasing top-to-bottom? [b]p4.[/b] Noah has an old-style M&M machine. Each time he puts a coin into the machine, he is equally likely to get $1$ M&M or $2$ M&M’s. He continues putting coins into the machine and collecting M&M’s until he has at least $6$ M&M’s. What is the probability that he actually ends up with $7$ M&M’s? [b]p5.[/b] Erik wants to divide the integers $1$ through $6$ into nonempty sets $A$ and $B$ such that no (nonempty) sum of elements in $A$ is a multiple of $7$ and no (nonempty) sum of elements in $B$ is a multiple of $7$. How many ways can he do this? (Interchanging $A$ and $B$ counts as a different solution.) [b]p6.[/b] A subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ of size $3$ is called special if whenever $a$ and $b$ are in the set, the remainder when $a + b$ is divided by $8$ is not in the set. ($a$ and $b$ can be the same.) How many special subsets exist? [b]p7.[/b] Let $F_1 = F_2 = 1$, and let $F_n = F_{n-1} + F_{n-2}$ for all $n \ge 3$. For each positive integer $n$, let $g(n)$ be the minimum possible value of $$|a_1F_1 + a_2F_2 + ...+ a_nF_n|,$$ where each $a_i$ is either $1$ or $-1$. Find $g(1) + g(2) +...+ g(100)$. [b]p8.[/b] Find the smallest positive integer $n$ with base-$10$ representation $\overline{1a_1a_2... a_k}$ such that $3n = \overline{a_1a_2


a_k1}$. [b]p9.[/b] How many ways are there to tile a $4 \times 6$ grid with $L$-shaped triominoes? (A triomino consists of three connected $1\times 1$ squares not all in a line.) [b]p10.[/b] Three friends want to share five (identical) muffins so that each friend ends up with the same total amount of muffin. Nobody likes small pieces of muffin, so the friends cut up and distribute the muffins in such a way that they maximize the size of the smallest muffin piece. What is the size of this smallest piece? [u]Numerical tiebreaker problems:[/u] [b]p11.[/b] $S$ is a set of positive integers with the following properties: (a) There are exactly 3 positive integers missing from $S$. (b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow $a$ and $b$ to be the same.) How many possibilities are there for the set $S$? [b]p12.[/b] In the trapezoid $ABCD$, both $\angle B$ and $\angle C$ are right angles, and all four sides of the trapezoid are tangent to the same circle. If $\overline{AB} = 13$ and $\overline{CD} = 33$, find the area of $ABCD$. [b]p13.[/b] Alice wishes to walk from the point $(0, 0)$ to the point $(6, 4)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0, 1)$ to the point $(6, 5)$ in increments of $(1, 0)$ and $(0,1)$. How many ways are there for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times)? [b]p14.[/b] The continuous function $f(x)$ satisfies $9f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y$. If $f(1) = 3$, what is $f(-3)$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?

Let $n\ge 3$ be an integer. Put $n^2$ cards, each labelled $1,2,\ldots ,n^2$ respectively, in any order into $n$ empty boxes such that there are exactly $n$ cards in each box. One can perform the following operation: one first selects $2$ boxes, takes out any $2$ cards from each of the selected boxes, and then return the cards to the other selected box. Prove that, for any initial order of the $n^2$ cards in the boxes, one can perform the operation finitely many times such that the labelled numbers in each box are consecutive integers.

Let $P$, $Q$, $R$ be points on the sides $AB$, $BC$, $CA$ of a triangle $ABC$ such that $AP=CQ$ and the quadrilateral $RPBQ$ is cyclic. The tangents to the circumcircle of triangle $ABC$ at the points $C$ and $A$ intersect the lines $RQ$ and $RP$ at the points $X$ and $Y$, respectively. Prove that $RX=RY$.

Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.

A set of points in the plane is called [i]sane[/i] if no three points are collinear and the angle between any three distinct points is a rational number of degrees. (a) Does there exist a countably infinite sane set $\mathcal{P}$? (b) Does there exist an uncountably infinite sane set $\mathcal{Q}$? [i]Proposed by Tony Wang[/i]

Given an equilateral triangle, all points on its sides are colored in one of two given colors. Prove that the is a right-angled triangle such that its three vertices are in the same color and on the sides of the equilateral triangle. [i]Alhaji Akbar, Jakarta[/i]

There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation: The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him. Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.

Find the sum of all positive two-digit integers that are divisible by each of their digits.

Find the remainder when $2020!$ is divided by $2020^2.$ [i]Proposed by Kevin Zhao[/i]