Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$

The equation $z^6 + z^3 + 1$ has one complex root with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$.

Let ${ABC}$ be a non-equilateral triangle. The incircle is tangent to the sides ${BC,CA,AB}$ at ${A_1,B_1,C_1}$, respectively, and M is the orthocenter of triangle ${A_1B_1C_1}$. Prove that ${M}$ lies on the line through the incenter and circumcenter of ${\vartriangle ABC}$.

Consider the set $X =\{1, 2,3, ...,2018\}$. How many positive integers $k$ with $2 \le k \le 2017$ that satisfy the following conditions: i) There exists some partition of the set $X$ into $1009$ disjoint pairs which are $(a_1, b_1),(a_2, b_2), ...,(a_{1009}, b_{1009})$ with $|a_i - b_i| \in \{1, k\}$. ii) For all partitions satisfy the condition (i), the sum $T = \sum^{1009}_{i=1} |a_i - b_i|$ has the right most digit is $9$

Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\cdots,10\}, \\ &\{11,12,13,\cdots,20\},\\ &\{21,22,23,\cdots,30\},\\ &\vdots\\ &\{991,992,993,\cdots,1000\}. \end{align*} How many of these sets contain exactly two multiples of $7$? $\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$

Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules: i) Any incantation can appear no more than once; ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it; iii)The first person who cannot spell an incantation loses the contest. Answer the following questions: a) If A says '$STAGEPREIMO$' first, then who will win? b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.

Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$. (a) Calculate $u_{1990}$. (b) Find the number of indices $n\le1990$ for which $u_n=0$. (c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$.

For a natural number $m>1$ we'll denote with $f(m)$ the sum of all natural numbers less than $m$, which are also coprime to $m$. Find all natural numbers $n$, such that there exist natural numbers $k$ and $\ell$ which satisfy $f(n^{k})=n^{\ell}$.

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet? $\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Joy has $33$ thin rods, one each of every integer length from $1$ cm through $30$ cm, and also three more rods with lengths $3$ cm, $7$ cm, and $15$ cm. She places those three rods on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? [i]Proposed by Michael Tang[/i]

The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$. [asy] unitsize(1 cm); pair A, B, C, D, O; D = (0,0); B = 3*dir(180 + 72); C = 3*dir(180 + 72 + 36); A = extension(D, D + (1,0), C, C + dir(180 - 36)); O = extension(A, C, B, D); draw(A--B--C--D--cycle); draw(B--D); draw(A--C); dot("$A$", A, N); dot("$B$", B, SW); dot("$C$", C, SE); dot("$D$", D, N); dot("$O$", O, E); [/asy] Attention: the figure is not drawn to scale.

Let $ \left(a_{n} \right)_{n \equal{} 1}^{\infty }$ be a sequence on real numbers such that $ a{}_{n \plus{} 1} \equal{} a_{n} a_{n \plus{} 2}$ for every $ n\ge 1$. The number of elements in the set $ \left\{a_{n} : n\ge 1\right\}$ cannot be $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$

Gilda has a bag of marbles. She gives $20 \%$ of them to her friend Pedro. The, Gilda gives $10 \%$ of what is left to her other friend, Ebony. Finally, Gilda gives $25 \%$ of what is left in the bag to her brother. What percentage of her original bag does she have left? $\textbf{(A) } 20 \qquad\textbf{(B) } 33\tfrac{1}{3} \qquad\textbf{(C) } 38 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 54$

Let $n\ge3$ be a positive integer. We say that a set $S$ of positive integers is good if $|S|=n$, no element of S is a multiple of n, and the sum of all elements of $S$ is not a multiple of $n$ either. Find, in terms of $n$, the least positive integer $d$ for which there exists a good set $S$ such that there are exactly d nonempty subsets of $S$ the sum of whose elements is a multiple of $n$. Proposed by Aleksandar Makelov, Burgas, Bulgaria and Nikolai Beluhov, Stara Zagora, Bulgaria

Noew is writing a $15$-problem mock AIME consisting of four subjects of problems: algebra, geometry, combinatorics, and number theory. The AIME is considered [i]somewhat evenly distributed[/i] if there is at least one problem of each subject and there are at least six combinatorics problems. Two AIMEs are considered [i]similar[/i] if they have the same subject distribution (same number of each subject). How many non-similar somewhat evenly distributed mock AIMEs can Noew write? [i]2019 CCA Math Bonanza Lightning Round #2.1[/i]

Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins. Which player has the winning strategy?

Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.

In the quadrilateral $ABCD$ the condition $AD = AB + CD$ is fulfilled. The bisectors of the angles $BAD$ and $ADC$ intersect at the point $P $, as shown in Fig. Prove that $BP = CP$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/67268635aaef9c6dc3363b00453b327cbc01f3.png[/img] (Maria Rozhkova)

Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\] [i]A. Mate[/i]

In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.

Let $a_1, a_2, \dots, a_{2019}$ be positive integers and $P$ a polynomial with integer coefficients such that, for every positive integer $n$, $$P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.$$ Prove that $P$ is a constant polynomial.

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube. [i]Dinu Serbanescu[/i]

What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).