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).

Find the graph of the function $y=x+|1-x^3|$.

What is the sum of $3+3^2+3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ in $\mod 11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 10 $

Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and \[a^3 + b^3 = p^n\mbox{.}\] [i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]

Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers? $\textbf{(A)}\mbox{ }360\qquad\textbf{(B)}\mbox{ }388\qquad\textbf{(C)}\mbox{ }412\qquad\textbf{(D)}\mbox{ }416\qquad\textbf{(E)}\mbox{ }424$

Given $n$ points, $n > 4$. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)

Let $T(p)$ denote the number of right triangles with integer side lengths and one of its side lengths being $p$. Which of the following values of $p$ produces the greatest possible value of $T(p)$ among all five answer choices? $\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }54$

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied \[f(x+y) + f(x)f(y) = f(xy) + 1 \] $\forall x, y \in \mathbb{R}$

For how many positive integers $n \le 1000$ does the equation in real numbers $x^{\lfloor x \rfloor } = n$ have a positive solution for $x$?

In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$. (RK Gordin)

Compute the number of five-digit positive integers $\overline{vwxyz}$ for which \[ (10v+w) + (10w+x) + (10x+y) + (10y+z) = 100. \][i]Proposed by Evan Chen[/i]

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

[list] [*]There are two semi-infinite plane mirrors inclined physically at a non-zero angle with the inner surfaces being reflective.\\ Prove that all lines of incident/reflected rays are tangential to a particular circle for any given incident ray being incident on a reflective side. Assume that the incident ray lies on one of the normal planes to the mirrors.[/*] [*] There's a cone of an arbitrary base with large enough length.\\ The inner surface polished (Outer surface is absorbing in nature) and the apex is fixed to a point. The cone is being rotated around the apex at an angular speed $\omega$ around the vertical axis and assume that a large part of the inside is visible horizontally. A fixed horizontal ray is projected from outside towards the cone (which often falls inside of it), prove that all the lines of incident ray/reflected rays at all instants lie tangential to a particular sphere.\\ Try guessing the radius of the sphere with the parameters you observe.[/*]

In a triangle $ABC$ point $I$ is the incenter, $I_A$ - excenter, $W$ - midpoint of the arc $BAC$ of circumcircle $\omega$ of $ABC$. Point $H$ is the projection of $I_A$ on $IW$. The tangent line to the circumcircle $BIC$ in point $I$ intersects $\omega$ in $E, F$. Prove that the perpendicular bisector to $AI$ is tangent to the circumcircle $EFH$ [i]M. Zorka[/i]