The football tournament was played in one round. $3$ points were given for a win, $1$ point for a draw, and $0$ points for a loss. Could it be that the first place team under the old scoring system (win - $2$ points, draw - $1$ point, loss - $0$) would be last?

Lizzie and Alex are playing a game on the whiteboard. Initially, $n$ twos are written on the board. On a player’s turn they must either 1. change any single positive number to 0, or 2. subtract one from any positive number of positive numbers on the board. The game ends once all numbers are 0, and the last player who made a move wins. If Lizzie always plays first, find all $n$ for which Lizzie has a winning strategy.

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

Show that for any $i=1,2,3$, there exist infinity many positive integer $n$, such that among $n$, $n+2$ and $n+28$, there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.

Find the probability such that when a polynomial in $\mathbb Z_{2027}[x]$ having degree at most $2026$ is chosen uniformly at random, $$x^{2027}-x | P^{k}(x) - x \iff 2021 | k $$ (note that $2027$ is prime). Here $P^k(x)$ denotes $P$ composed with itself $k$ times. [i]Proposed by Grant Yu[/i]

Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$

Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.

Consider a multiplicative function \( f \) from the positive integers to the unit disk centered at the origin, that is, \( f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} \) such that \( f(mn) = f(m)f(n) \). Prove that for every \( \epsilon > 0 \) and every integer \( k > 0 \), there exist \( k \) distinct positive integers \( a_1, a_2, \dots, a_k \) such that \( \text{gcd}(a_1, a_2, \dots, a_k) = k \) and \( d(f(a_i), f(a_j)) < \epsilon \) for all \( i, j = 1, \dots, k \).

Find all the sequences of natural $k_n$ with two properties: a) $k_n \le n \sqrt {n}$ for all $n$ b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$ where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$

Solve the system of equations in integers $$x + y + z = 3$$ $$x^3 + y^3 + z^3 = 3$$

Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.

Let $N_{15}$ be the answer to problem 15. For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.

Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

Positive integers are written on the blackboard one after another. The next integer $a_{n+1}$ (to be written after $a_1$,$a_2$,$...$,$a_n$) is an arbitrary integer not representable as a sum of several previous integers taken one or more times (i.e. $a_{n+1}$ is not of the form $k_1 *a_i + k_2 *a_2 + ... + k_n *a_n$ where$ k_1$, $k_2$,$..$, $k_n$ are non-negative integers). Prove that the process of writing cannot be infinite. (A Belov)

Denote by $ d(n)$ the number of all positive divisors of a natural number $ n$ (including $ 1$ and $ n$). Prove that there are infinitely many $ n$, such that $ n/d(n)$ is an integer.

Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3\plus{}b^3\equal{}c^3\plus{}d^3$ and $ a\plus{}b\equal{}c\plus{}d$.

A positive integer $N$ is [i]rioplatense[/i] if it satifies the following conditions: 1 -There exist $34$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. 2 - There [b]not[/b] exist $30$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. Determine all rioplatense numbers.

There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]B16 / G11[/b] Let triangle $ABC$ be an equilateral triangle with side length $6$. If point $D$ is on $AB$ and point $E$ is on $BC$, find the minimum possible value of $AD + DE + CE$. [b]B17 / G12[/b] Find the smallest positive integer $n$ with at least seven divisors. [b]B18 / G13[/b] Square $A$ is inscribed in a circle. The circle is inscribed in Square $B$. If the circle has a radius of $10$, what is the ratio between a side length of Square $A$ and a side length of Square $B$? [b]B19 / G14[/b] Billy Bob has $5$ distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other? [b]B20 / G15[/b] Six people make statements as follows: Person $1$ says “At least one of us is lying.” Person $2$ says “At least two of us are lying.” Person $3$ says “At least three of us are lying.” Person $4$ says “At least four of us are lying.” Person $5$ says “At least five of us are lying.” Person $6$ says “At least six of us are lying.” How many are lying? [u]Set 5[/u] [b]B21 / G16[/b] If $x$ and $y$ are between $0$ and $1$, find the ordered pair $(x, y)$ which maximizes $(xy)^2(x^2 - y^2)$. [b]B22 / G17[/b] If I take all my money and divide it into $12$ piles, I have $10$ dollars left. If I take all my money and divide it into $13$ piles, I have $11$ dollars left. If I take all my money and divide it into $14$ piles, I have $12$ dollars left. What’s the least amount of money I could have? [b]B23 / G18[/b] A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation. [b]B24 / G20[/b] A regular $12$-sided polygon is inscribed in a circle. Gaz then chooses $3$ vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right? [b]B25 / G22[/b] A book has at most $7$ chapters, and each chapter is either $3$ pages long or has a length that is a power of $2$ (including $1$). What is the least positive integer $n$ for which the book cannot have $n$ pages? [u]Set 6[/u] [b]B26 / G26[/b] What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers? [b]B27 / G27[/b] Estimate $12345^{\frac{1}{123}}$. [b]B28 / G28[/b] Let $O$ be the center of a circle $\omega$ with radius $3$. Let $A, B, C$ be randomly selected on $\omega$. Let $M$, $N$ be the midpoints of sides $BC$, $CA$, and let $AM$, $BN$ intersect at $G$. What is the probability that $OG \le 1$? [b]B29 / G29[/b] Let $r(a, b)$ be the remainder when $a$ is divided by $b$. What is $\sum^{100}_{i=1} r(2^i , i)$? [b]B30 / G30[/b] Bongo flips $2023$ coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets $HHHT T HT T HHHHT H$, he’d have maximal runs of length $3, 1, 4, 1$. Bongo squares the lengths of all his maximal runs and adds them to get a number $M$. What is the expected value of $M$? - - - - - - [b]G19[/b] Let $ABCD$ be a square of side length $2$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. Let the intersection of $BN$ and $CM$ be $E$. Find the area of quadrilateral $NECD$. [b]G21[/b] Quadrilateral $ABCD$ has $\angle A = \angle D = 60^o$. If $AB = 8$, $CD = 10$, and $BC = 3$, what is length $AD$? [b]G23[/b] $\vartriangle ABC$ is an equilateral triangle of side length $x$. Three unit circles $\omega_A$, $\omega_B$, and $\omega_C$ lie in the plane such that $\omega_A$ passes through $A$ while $\omega_B$ and $\omega_C$ are centered at $B$ and $C$, respectively. Given that $\omega_A$ is externally tangent to both $\omega_B$ and $\omega_C$, and the center of $\omega_A$ is between point $A$ and line $\overline{BC}$, find $x$. [b]G24[/b] For some integers $n$, the quadratic function $f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12)$ has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form $2^k$ for some nonnegative integer $k$. What is the sum of all possible values of $n$? [b]G25[/b] In a triangle, let the altitudes concur at $H$. Given that $AH = 30$, $BH = 14$, and the circumradius is $25$, calculate $CH$ PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1[/b] What is the sum of the first $5$ positive integers? [b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$? [b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number? [b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen? [b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$? [b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle? [b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus? [b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win? [b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece? [b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$? [b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime? Note: $1$ is not prime. [b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement? [b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares? [b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions? [b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card? [b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$? [b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$? [b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$? [b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$. [b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

[color=darkblue]Let $ M$ be a set of aritmetic progressions with integer terms and ratio bigger than $ 1$. [b]a)[/b] Prove that the set of the integers $ \mathbb{Z}$ can be written as union of the finite number of the progessions from $ M$ with different ratios. [b]b)[/b] Prove that the set of the integers $ \mathbb{Z}$ can not be written as union of the finite number of the progessions from $ M$ with ratios integer numbers, any two of them coprime.[/color]