Let $m\ge2$ be an integer. The sequence $(a_n)_{n\in\mathbb N}$ is defined by $a_0=0$ and $a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor}$ for all $n$. Determine $\lim_{n\to\infty}\frac{a_n}n$.

$25$ checkers are placed on $25$ leftmost squares of $1 \times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such that all the checkers could be placed in the row of $25$ successive squares but in the reverse order.

* What is the radius of the largest possible circle inscribed into a cube with side $a$?

Cards numbered with numbers $1$ to $1000$ are to be placed on the cells of a $1\times 1994$ rectangular board one by one, according to the following rule: If the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be put on it. Prove that the number of possible arrangements is not more than half a mllion.

Call a quadratic [i]invasive[/i] if it has $2$ distinct real roots. Let $P$ be a quadratic polynomial with real coefficients. Prove that $P(x)$ is invasive [b]if and only if[/b] there exists a real number $c \neq 0$ such that $P(x) + P(x - c)$ is invasive.

Find all real polynomials $ p(x)$ of degree $ n \ge 2$ for which there exist real numbers $ r_1 < r_2 < ... < r_n$ such that (i) $ p(r_i) \equal{} 0, 1 \le i \le n$, and (ii) $ p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1$. [b]Follow-up:[/b] In terms of $ n$, what is the maximum value of $ k$ for which $ k$ consecutive real roots of a polynomial $ p(x)$ of degree $ n$ can have this property? (By "consecutive" I mean we order the real roots of $ p(x)$ and ignore the complex roots.) In particular, is $ k \equal{} n \minus{} 1$ possible for $ n \ge 3$?

The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$.

The diagonals of a cyclic convex quadrilateral $ABCD$ intersect at $P$. A circle through $P$ touches the side $CD$ at its midpoint $M$ and intersects the segments $BD$ and $AC$ again at the points $Q$ and $R$ respectively. Let $S$ be the point on segment $BD$ such that $BS = DQ$. The line through $S$ parallel to $AB$ intersects $AC$ at $T$. Prove that $AT = RC$.

As everyone knows, the people of [i]Plane Land[/i] love Planimetrics. Therefore, they imagine their country as completely planar, every city in the country as a geometric point and every road as the line segment connecting two points. Additionally to the existing cities, it is possible to build [i]roundabouts[/i], i.e. points in the road network from where at least two roads emanate. All road crossings or junctions are build as roundabouts. Via this route network, every two cities should be connected by a sequence of roads and possibly roundabouts. In Plane Land, the length of a road is taken as the geometric length of the corresponding line segment. The ingenious road engineer Armin Asphalt presents a new road map, of which it is known that there is no road network with a smaller total length of all roads. Moreover, there is no road map with the same total length of all roads and fewer roundabouts. Prove that in the road map of Armin Asphalt, at most three roads emanate from each city, and exactly three from each roundabout.

Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.

Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$, $BC=5$, and $CD=7$.

A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?

Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.

For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.

Let $A, B, C, D$ be points in a line $l$ in this order where $AB = BC$ and $AC = CD$. Let $w$ be a circle that passes in the points $B$ and $D$, a line that passes by $A$ intersects $w$ in the points $P$ and $Q$(the point $Q$ is in the segment $AP$). Let $M$ be the midpoint of $PD$ and $R$ is the symmetric of $Q$ by the line $l$, suppose that the segments $PR$ and $MB$ intersect in the point $N$. Prove that the quadrilateral $PMNC$ is cyclic

Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive? (4 points) Boris Frenkin

A $100 \times 100$ grid is given as shown. We choose a certain number of cells such that exactly two cells in each row and column are selected. Find the sum of numbers in these cells.

Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.

Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that: $ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

Determine all positive integers $n$ such that $ xy+1 \equiv 0 \; \pmod{n} $ implies that $ x+y \equiv 0 \; \pmod{n}$.

3 countries $A, B, C$ participate in a competition where each country has 9 representatives. The rules are as follows: every round of competition is between 1 competitor each from 2 countries. The winner plays in the next round, while the loser is knocked out. The remaining country will then send a representative to take on the winner of the previous round. The competition begins with $A$ and $B$ sending a competitor each. If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out. The remaining team is the champion. [b]I.[/b] At least how many games does the champion team win? [b]II.[/b] If the champion team won 11 matches, at least how many matches were played?

[u]Round 5 [/u] [b]p13.[/b] Circles $\omega_1$, $\omega_2$, and $\omega_3$ have radii $8$, $5$, and $5$, respectively, and each is externally tangent to the other two. Circle $\omega_4$ is internally tangent to $\omega_1$, $\omega_2$, and $\omega_3$, and circle $\omega_5$ is externally tangent to the same three circles. Find the product of the radii of $\omega_4$ and $\omega_5$. [b]p14.[/b] Pythagoras has a regular pentagon with area $1$. He connects each pair of non-adjacent vertices with a line segment, which divides the pentagon into ten triangular regions and one pentagonal region. He colors in all of the obtuse triangles. He then repeats this process using the smaller pentagon. If he continues this process an infinite number of times, what is the total area that he colors in? Please rationalize the denominator of your answer. p15. Maisy arranges $61$ ordinary yellow tennis balls and $3$ special purple tennis balls into a $4 \times 4 \times 4$ cube. (All tennis balls are the same size.) If she chooses the tennis balls’ positions in the cube randomly, what is the probability that no two purple tennis balls are touching? [u]Round 6 [/u] [b]p16.[/b] Points $A, B, C$, and $D$ lie on a line (in that order), and $\vartriangle BCE$ is isosceles with $\overline{BE} = \overline{CE}$. Furthermore, $F$ lies on $\overline{BE}$ and $G$ lies on $\overline{CE}$ such that $\vartriangle BFD$ and $\vartriangle CGA$ are both congruent to $\vartriangle BCE$. Let $H$ be the intersection of $\overline{DF}$ and $\overline{AG}$, and let $I$ be the intersection of $\overline{BE}$ and $\overline{AG}$. If $m \angle BCE = arcsin \left( \frac{12}{13} \right)$, what is $\frac{\overline{HI}}{\overline{FI}}$ ? [b]p17.[/b] Three states are said to form a tri-state area if each state borders the other two. What is the maximum possible number of tri-state areas in a country with fifty states? Note that states must be contiguous and that states touching only at “corners” do not count as bordering. [b]p18.[/b] Let $a, b, c, d$, and $e$ be integers satisfying $$2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c +\sqrt[3]{2}d + e = 0$$ and $$25\sqrt5 i + 25a - 5\sqrt5 ib - 5c + \sqrt5 id + e = 0$$ where $i =\sqrt{-1}$. Find $|a + b + c + d + e|$. [u]Round 7[/u] [b]p19.[/b] What is the greatest number of regions that $100$ ellipses can divide the plane into? Include the unbounded region. [b]p20.[/b] All of the faces of the convex polyhedron $P$ are congruent isosceles (but NOT equilateral) triangles that meet in such a way that each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. (An isosceles triangle has two base angles and one vertex angle.) Find the sum of the numbers of faces, edges, and vertices of $P$. [b]p21.[/b] Find the number of ordered $2018$-tuples of integers $(x_1, x_2, .... x_{2018})$, where each integer is between $-2018^2$ and $2018^2$ (inclusive), satisfying $$6(1x_1 + 2x_2 +...· + 2018x_{2018})^2 \ge (2018)(2019)(4037)(x^2_1 + x^2_2 + ... + x^2_{2018}).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2784936p24472982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The probability that a set of three distinct vertices chosen at random from among the vertices of a regular $n$-gon determine an obtuse triangle is $\tfrac{93}{125}$. Find the sum of all possible values of $n$.

Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer $ n $ there exists a triangle of area $ n $ having its vertices of the chosen colour.