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]

Let $C$ be a variable interior point of a fixed segment $AB$. Equilateral triangles $ACB' $ and $CBA'$ are constructed on the same side and $ABC' $ on the other side of the line $AB$. (a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$. (b) Find the locus of $P$ as $C$ varies. (c) Prove that the centers $A'' ,B'' ,C''$ of the three triangles form an equilateral triangle. (d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties: 1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$. 2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$. [i]Proposed by Pavel Ciurea[/i]

Let $a$ and $b$ be two positive real numbers, and $AB$ a segment of length $a$ on a plane. Let $C,D$ be two variable points on the plane such that $ABCD$ is a non-degenerate convex quadrilateral with $BC=CD=b$ and $DA=a$. It is easy to see that there is a circle tangent to all four sides of the quadrilateral $ABCD$. Find the precise locus of the point $I$.

A number is called [i]flippy[/i] if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer. [i]Proposed by Evan Chen[/i]

Let $ r_1 $, $ r_2 $, $ r_3 $ be the (possibly complex) roots of the polynomial $ x^3 + ax^2 + bx + \dfrac{4}{3} $. How many pairs of integers $ a $, $ b $ exist such that $ r_1^3 + r_2^3 + r_3^3 = 0 $?

Let $ABC$ be a triangle. On the sides $BC$, $CA$, $AB$ of the triangle, construct outwardly three squares with centres $O_a$, $O_b$, $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$. Given that $A$ lies on $\omega$, prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$. [i]Sam Bealing, United Kingdom[/i]

Given is a triangle $ABC$ with circumcenter $O$. Points $D, E$ are chosen on the angle bisector of $\angle ABC$ such that $EA=EB, DB=DC$. If $P, Q$ are the circumcenters of $(AOE), (COD)$, prove that either the line $PQ$ coincides with $AC$ or $PQCA$ is cyclic.

For positive reals $a,b$ and $c$ with $ab+bc+ca=1$, show that \[\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}\] [i]Proposed by Morteza Saghafian[/i]

In $\vartriangle ABC, D, E, F$ are the midpoints of $AB, BC, CA$ respectively. Denote by $O_A, O_B, O_C$ the incenters of $\vartriangle ADF, \vartriangle BED, \vartriangle CFE$ respectively. Prove that $O_AE, O_BF, O_CD$ are concurrent.