[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

Given an equilateral triangle lattice composed of $\frac{n(n+1)}{2}$ points (as shown in the figure), record the number of equilateral triangles with three points in the lattice as vertices as $f(n)$. Find an expression for $f(n)$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/1de27231e8ef9c1c6a3dfd590a7c71adc508d6.png[/img]

There are $100$ people in a queue waiting to enter a hall. The hall has exactly $100$ seats numbered from $1$ to $100$. The first person in the queue enters the hall, chooses any seat and sits there. The $n$-th person in the queue, where $n$ can be $2, . . . , 100$, enters the hall after $(n-1)$-th person is seated. He sits in seat number $n$ if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which $100$ seats can be filled up, provided the $100$-th person occupies seat number $100$.

A [i]figuric [/i] is a convex polyhedron with $26^{5^{2019}}$ faces. On every face of a figuric we write down a number. When we throw two figurics (who don't necessarily have the same set of numbers on their sides) into the air, the figuric which falls on a side with the greater number wins; if this number is equal for both figurics, we repeat this process until we obtain a winner. Assume that a figuric has an equal probability of falling on any face. We say that one figuric rules over another if when throwing these figurics into the air, it has a strictly greater probability to win than the other figuric (it can be possible that given two figurics, no figuric rules over the other). Milisav and Milojka both have a blank figuric. Milisav writes some (not necessarily distinct) positive integers on the faces of his figuric so that they sum up to $27^{5^{2019}}$. After this, Milojka also writes positive integers on the faces of her figuric so that they sum up to $27^{5^{2019}}$. Is it always possible for Milojka to create a figuric that rules over Milisav's? [i]Proposed by Bojan Basic[/i]

One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as it is. [list=a] [*] Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end. [*] Is it possible that precisely one quarter of the glasses contain green liquid at the end? [/list]

Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus. [i]Proposed by Tommy Walker Mackay, United Kingdom[/i]

The base of a prism is an $n$-gon. We wish to colour its $2n$ vertices in three colours in such a way that every vertex is connected by edges to vertices of all three colours. (a) Prove that if $n$ is divisible by $3$, then the task is possible. {b) Prove that if the task is possible, then $n$ is divisible by $3$. (A Shapovalov)

Let $f(k)$ denote the number of triples $(a, b, c)$ of positive integers satisfying $a + b + c = 2020$ with $(k - 1)$ not dividing $a, k$ not dividing $b$, and $(k + 1)$ not dividing $c$. Find the product of all integers $k$ in the range 3 \le k \le 20 such that $(k + 1)$ divides $f(k)$.

[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $. For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$. Find the number of all [i]palindromic partitions[/i] of the number $2006$.

Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infinity, what is the ratio between the number of triangles and the number of squares?

The game Battleship is played on a $10\times10$ grid. A [i]fleet[/i] consists of 10 ships: one occupying $4$ cells, two occupying $3$ cells each, three occupying $2$ cells each and four occupying $1$ cell each (see figure). [asy] size(10cm); // Function to draw a square centered at a given position void drawSquare(pair center, real sideLength) { real halfSide = sideLength / 2; draw(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide))); } // Side length of each square real sideLength = 1; // Coordinates for the squares pair[] positions = { // Top row remains the same (0, 0), (1, 0), (3, 0), (4, 0), (6, 0), (7, 0), (9, 0), (11, 0), (13, 0), (15, 0), // Bottom row moved one square (1 unit) to the right (2, 2), (3, 2), (4, 2), (5, 2), (7, 2), (8, 2), (9, 2), (11, 2), (12, 2), (13, 2) }; // Draw all squares for (pair pos : positions) { drawSquare(pos, sideLength); } [/asy] Ships can be placed either horizontally or vertically, but they must not touch each other, not even at a vertex. Is it possible to place two fleets on the same board according to these rules?

Given two positive integers $n$ and $k$, we say that $k$ is [i]$n$-ergetic[/i] if: However the elements of $M=\{1,2,\ldots, k\}$ are coloured in red and green, there exist $n$ not necessarily distinct integers of the same colour whose sum is again an element of $M$ of the same colour. For each positive integer $n$, determine the least $n$-ergetic integer, if it exists.

Let $S = \mathbb Z \times \mathbb Z$. A subset $P$ of $S$ is called [i]nice[/i] if [list] [*] $(a, b) \in P \implies (b, a) \in P$ [*] $(a, b)$, $(c, d)\in P \implies (a + c, b - d) \in P$ [/list] Find all $(p, q) \in S$ so that if $(p, q) \in P$ for some [i]nice[/i] set $P$ then $P = S$.

Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.

A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.

Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$. Construct an $n \times 3$ table where the entries of the $k$-th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$. Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table. [list=a] [*] Prove that at least three entries remain at the end of the procedure. [*] Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain. [/list]

At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. Call a person [i]junior[/i] if they are younger than all their friends, and [i]senior[/i] if they are older than all their friends. A person with no friends is both [i]junior[/i] and [i]senior[/i]. A sequence of pairwise distinct people $A_1, \dots, A_m$ is called [i]photogenic[/i] if: 1. $A_1$ is [i]junior[/i], 2. $A_m$ is [i]senior[/i], and 3. $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \leq i \leq m-1$. Let $k$ be a positive integer such that for every [i]photogenic[/i] sequence $A_1, \dots, A_m$, $m$ is not divisible by $k$. Prove that the people at the party can be partitioned into $k$ groups so that no two people in the same group are friends. [i]Proposed by Shantanu Nene[/i]

Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

In the every cell of the board $5\times5$ there is one of the numbers: $-1$, $0$, $1$. It is true that in every $2 \times 2$ square there are three numbers summing up to $0$. Determine the maximal sum of all numbers in a board.

Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?

What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?

Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$, and turns over the coins next to it to the left and to the right (if any). Determine all $n$ such that Jack can make all coins show $T$ after a finite number of operations.

Initially, the number $10$ is written on the board. In each subsequent moves, you can either (i) erase the number $1$ and replace it with a $10$, or (ii) erase the number $10$ and replace it with a $1$ and a $25$ or (iii) erase a $25$ and replace it with two $10$. After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?