Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$. [i]V. Kamianetski[/i]

A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ [b]a)[/b] Demonstrate that there are no injective functions with this property. [b]b)[/b] Do exist surjective functions having this property?

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? $\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$

How many handles has the Riemann surface of the function \[w=\sqrt{1+z^n}\]

Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.

Suppose that a triangle $ABC$ with incenter $I$ satisfies $CA+AI=BC$. Find the ratio between the measures of the angles $\angle BAC$ and $\angle CBA$.

Prove that for any positive integer $n$ there exists a matrix of the form $$A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix},$$ (a) with nonzero entries, (b) with positive entries, such that the entries of $A^n$ are all perfect squares.

Let $n$ be a fixed positive integer. Ana and Banana are playing a game. First, Ana picks a subset $S$ of $\{1,2,\ldots,n\}$. Then for each $k=1,2,\ldots,n$, she tells Banana how many numbers from $k-1$ to $k+1$ she has picked (i.e. $\lvert S \cap \{k-1,k,k+1\}\rvert$). Then Banana guesses $S$; she wins if her guess is correct and she loses otherwise. (a) Determine all $n$ for which Banana will win regardless of what Ana chooses. (b) For the values of $n$ for which Ana can win, determine the number of sets $S$ she can choose so as to do so.

Let $ABCDEFG$ be a regular heptagon. We'll call the sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$ and $GA$ opposite to the vertices $E$, $F$, $G$, $A$, $B$, $C$ and $D$ respectively. If $M$ is a point inside the heptagon, we'll say that the line through $M$ and a vertex of the heptagon intersects a side of it (without the vertices) at a $\textit{perfect}$ point, if this side is opposite the vertex. Prove that for every choice of $M$, the number of $\textit{perfect}$ points is always odd.

Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below: [asy] size(4.5cm); label("$1$", (0,0)); label("$1$", (-0.5,-2/3)); label("$1$", (0.5,-2/3)); label("$1$", (-1,-4/3)); label("$3$", (0,-4/3)); label("$1$", (1,-4/3)); label("$1$", (-1.5,-2)); label("$5$", (-0.5,-2)); label("$5$", (0.5,-2)); label("$1$", (1.5,-2)); label("$1$", (-2,-8/3)); label("$7$", (-1,-8/3)); label("$11$", (0,-8/3)); label("$7$", (1,-8/3)); label("$1$", (2,-8/3)); [/asy] Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$

Every two of $200$ points in space are connected by a segment, no two intersecting each other. Each segment is painted in one colour, and the total number of colours is $k$. Peter wants to paint each of the $200$ points in one of the colours used to paint the segments, so that no segment connects two points both in the same colour as the segment itself. Can Peter always do this if (a) k = 7; [i](4 points)[/i] (b) k = 10? [i](4 points)[/i]

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

[b]p1.[/b] $17.5\%$ of what number is $4.5\%$ of $28000$? [b]p2.[/b] Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$. The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] In the $xy$-plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$. Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at? [b]p4.[/b] What are the last two digits of the sum of the first $2021$ positive integers? [b]p5.[/b] A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] How many terms are in the arithmetic sequence $3$, $11$, $...$, $779$? [b]p7.[/b] Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$? [b]p8.[/b] What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$? [b]p9.[/b] Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$, $AB = 10$, $BC = 9$, and the area of $\vartriangle ABC$ is $36$. Compute the length of $AC$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/b648d0d60c186d01493fcb4e21b5260c46606e.png[/img] [b]p10.[/b] If $x + y - xy = 4$, and $x$ and $y$ are integers, compute the sum of all possible values of$ x + y$. [b]p11.[/b] What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap? [b]p12.[/b] $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$. Compute the smallest possible value of $N$. [b]p13.[/b] Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years? [b]p14.[/b] Say there is $1$ rabbit on day $1$. After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$? [b]15.[/b] Ajit draws a picture of a regular $63$-sided polygon, a regular $91$-sided polygon, and a regular $105$-sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have? [b]p16.[/b] Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p17.[/b] Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$'s adjacent. [b]p18.[/b] From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$, where $a$ and $b$ are positive integers. Compute $a + b$. [b]p19.[/b] Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$. He starts by putting the first marble in bucket $1$, the second marble in bucket $2$, the third marble in bucket $3$, etc. After placing a marble in bucket $9$, he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag? [img]https://cdn.artofproblemsolving.com/attachments/9/8/4c6b1bd07367101233385b3ffebc5e0abba596.png[/img] [b]p20.[/b] What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid? [hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]

Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.) What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)

The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x^2 + y) \ge (\frac{1}{x} + 1)f(y)$$ holds for all $x \in \mathbb{R} \setminus \{0\}$ and all $y \in \mathbb{R}$.

Arnaldo and Bernardo play a Super Naval Battle. Each has a board $n \times n$. Arnaldo puts boats on his board (at least one but not known how many). Each boat occupies the $n$ houses of a line or a column and the boats they can not overlap or have a common side. Bernardo marks $m$ houses (representing shots) on your board. After Bernardo marked the houses, Arnaldo says which of them correspond to positions occupied by ships. Bernardo wins, and then discovers the positions of all Arnaldo's boats. Determine the lowest value of $m$ for which Bernardo can guarantee his victory.

Solve the inequation $\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$.

For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

Let $n \ge 2$ be an integer. In each cell of a $4n \times 4n$ table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors $n^2$ moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

Each of integers from $1$ to $20$ are placed into the dots below. Two dots are [i]adjacent[/i], if below figure contains a line segment connecting them. Prove that how the numbers are arranged, it is possible to find an adjacent pair such that the difference between the numbers written on them is greater than $3$. [asy] real u=0.25cm; for(int i = 0; i < 4; ++i) { real v = u*(i+1); pair P1 = dir(90+0*72)*(0,v); pair P2 = dir(90+1*72)*(0,v); pair P3 = dir(90+2*72)*(0,v); pair P4 = dir(90+3*72)*(0,v); pair P5 = dir(90+4*72)*(0,v); dot(P1);dot(P2); dot(P3);dot(P4);dot(P5); path p = P1--P2--P3--P4--P5--cycle; draw(p); } [/asy]

A polygon is given in which any two adjacent sides are perpendicular. We call its two vertices non-friendly if the bisectors of the polygon emerging from these vertices are perpendicular. Prove that for any vertex the number of vertices that are not friends with it is even.

The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$, while its charge is $\frac12$ at pH $9.6$. Charge increases linearly with pH. What is the isoelectric point of glycine?