Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions: (1) $f(0) = 0, f(1) = 1,$ (2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$ Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$

Consider the set $ S$ of all numbers of the form $ a(n)\equal{}n^2\plus{}n\plus{}1, n \in \mathbb{N}.$ Show that the product $ a(n)a(n\plus{}1)$ is in $ S$ for all $ n \in \mathbb{N}$ and give an example of two elements $ s,t$ of $ S$ such that $ s,t \notin S$.

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

[b]6.1 / 7.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. [b]6.2 [/b] The natural numbers are arranged in a $3 \times 3$ table. Kolya and Petya crossed out 4 numbers each. It turned out that the sum of the numbers crossed out by Petya is three times the sum numbers crossed out by Kolya. What number is left uncrossed? $$\begin{tabular}{|c|c|c|}\hline 4 & 12 & 8 \\ \hline 13 & 24 & 14 \\ \hline 7 & 5 & 23 \\ \hline \end{tabular} $$ [b]6.3 [/b] Misha and Sasha left at noon on bicycles from city A to city B. At the same time, I left from B to A Vanya. All three travel at constant but different speeds. At one o'clock Sasha was exactly in the middle between Misha and Vanya, and at half past one Vanya was in the middle between Misha and Sasha. When Misha will be exactly in the middle between Sasha and Vanya? [b]6.4[/b] There are $35$ piles of nuts on the table. Allowed to add one nut at a time to any $23$ piles. Prove that by repeating this operation, you can equalize all the heaps. [b]6.5[/b] There are $64$ vertical stripes on the round drum, and each stripe you need to write down a six-digit number from digits $1$ and $2$ so that all the numbers were different and any two adjacent ones differed in exactly one discharge. How to do this? [b]6.6 / 7.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

Find all four digit positive integers such that the sum of the squares of the digits equals twice the sum of the digits.

Let $a>2$, $n>1$ integers such that $a^n-2^n$ is a perfect square. Prove that $a$ is a even number.

Among all triples $(a,b,c)$ of natural numbers satisfying \[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\] determine the one with the maximal value of $a$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D.16 / L.6[/b] Alex has $100$ Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most $1$ spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them? [b]D.17[/b] I start with the number $1$ in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket? [b]D.18 / L.12[/b] Point $P$ inside square $ABCD$ is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area $25$, $25$, and $15$, what are all the possible values for the area of the fourth triangle? [b]D.19[/b] Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at $100$ or more decibels, then they lose the game. Mr. Stein yells first, at $88$ decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win? [b]D.20 / L.15[/b] A semicircle of radius $1$ has line $\ell$ along its base and is tangent to line $m$. Let $r$ be the radius of the largest circle tangent to $\ell$, $m$, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of $r$ is the interval $[a, b]$. Find $b - a$. [u]Set 5[/u] [b]D.21 / L.14[/b] Hungryman starts at the tile labeled “$S$”. On each move, he moves $1$ unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats. [img]https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png [/img] [b]D.22 / L.11[/b] How many triples of nonnegative integers $(x, y, z)$ satisfy the equation $6x + 10y +15z = 300$? [b]D.23 / L.16[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]D.24 / L.13[/b] Tse and Cho are playing a game. Cho chooses a number $x \in [0, 1]$ uniformly at random, and Tse guesses the value of $x(1 - x)$. Tse wins if his guess is at most $\frac{1}{50}$ away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins? [b]D.25 / L.20[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.$$ [u]Set 6[/u] [i]This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.[/i] [b]D.26 / L.26[/b] What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were $47$ volunteers; this is the fifth MBMT. [b]D.27 / L.27[/b] William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than $10\%$ of the entire chocolate bar. [b]D.28 / L.28[/b] Let $x$ be the positive solution to the equation $x^{x^{x^x}}= 1.1$. Estimate $\frac{1}{x-1}$. [b]D.29 / L.29[/b] Estimate the number of dots in the following box: [img]https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png[/img] It may be useful to know that this image was produced by plotting $(4\sqrt{x}, y)$ some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval $[0, 1]$. [b]D.30 / L.30[/b] For a positive integer $n$, let $f(n)$ be the smallest prime greater than or equal to $n$. Estimate $$(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).$$ For $26 \le i \le 30$, let $E_i$ be your team’s answer to problem $i$ and let $A_i$ be the actual answer to problem $i$. Your score $S_i$ for problem $i$ is given by $S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5)$ $S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|)$ $S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|))$ $S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right)$ $S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)$ PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$. Note: $[r]$ denotes the integer part of $r$.

Find all prime numbers $ a,b,c$ that fulfill the equality: $ (a\minus{}2)!\plus{}2b!\equal{}22c\minus{}1$

Prove that the product of the 99 numbers $ \frac{k^3\minus{}1}{k^3\plus{}1},k\equal{}2,3,\ldots,100$ is greater than $ 2/3$.

A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.

In the number puzzle below, each cell contains a digit, each cell in the same bolded region has the same digit, and cells in different bolded regions have different digits. The answers to the clues are to be read as three-, four-, or five-digit numbers. Find the unique solution to the puzzle, given that no answer to any clue has a leading $0$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/23514673819aea46c30fd2947f8c82710a1fb3.png[/img]

Find all integers $n$ for which $n^7-41$ is the square of an integer

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Find all pairs $(a,b)$ of positive integers, such that for [b]every[/b] $n$ positive integer, the equality $a^n+b^n=c_n^{n+1}$ is true, for some $c_n$ positive integer.

Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.

[i]Each problem in this section will depend on the previous one! The values $A, B, C$, and $D$ refer to the answers to problems $1, 2, 3$, and $4$, respectively.[/i] [b]TR1.[/b] The number $2020$ has three different prime factors. What is their sum? [b]TR2.[/b] Let $A$ be the answer to the previous problem. Suppose$ ABC$ is a triangle with $AB = 81$, $BC = A$, and $\angle ABC = 90^o$. Let $D$ be the midpoint of $BC$. The perimeter of $\vartriangle CAD$ can be written as $x + y\sqrt{z}$, where $x, y$, and $z$ are positive integers and $z$ is not divisible by the square of any prime. What is $x + y$? [b]TR3.[/b] Let $B$ the answer to the previous problem. What is the unique real value of $k$ such that the parabola $y = Bx^2 + k$ and the line $y = kx + B$ are tangent? [b]TR4.[/b] Let $C$ be the answer to the previous problem. How many ordered triples of positive integers $(a, b, c)$ are there such that $gcd(a, b) = gcd(b, c) = 1$ and $abc = C$? [b]TR5.[/b] Let $D$ be the answer to the previous problem. Let $ABCD$ be a square with side length $D$ and circumcircle $\omega$. Denote points $C'$ and $D'$ as the reflections over line $AB$ of $C$ and $D$ respectively. Let $P$ and $Q$ be the points on $\omega$, with$ A$ and $P$ on opposite sides of line $BC$ and $B$ and $Q$ on opposite sides of line $AD$, such that lines $C'P$ and $D'Q$ are both tangent to $\omega$. If the lines $AP$ and $BQ$ intersect at $T$, what is the area of $\vartriangle CDT$? PS. You had better use hide for answers.

a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. [i]Moldavia Olympiad, 1994[/i]

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\] [PS: The original problem was this: Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\] But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]

[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten? [b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le 400$ that satisfy the following: $\bullet$ $n$ is a square number. $\bullet$ $n$ is one more than a multiple of $5$. $\bullet$ $n$ is even. [b]p3.[/b] How many positive integers less than $2019$ are either a perfect cube or a perfect square but not both? [b]p4.[/b] Felicia draws the heart-shaped figure $GOAT$ that is made of two semicircles of equal area and an equilateral triangle, as shown below. If $GO = 2$, what is the area of the figure? [img]https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png[/img] [b]p5.[/b] For distinct digits $A, B$, and $ C$: $$\begin{tabular}{cccc} & A & A \\ & B & B \\ + & C & C \\ \hline A & B & C \\ \end{tabular}$$ Compute $A \cdot B \cdot C$. [b]p6 [/b] What is the difference between the largest and smallest value for $lcm(a,b,c)$, where $a,b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive? [b]p7.[/b] Let $A$ and $B$ be points on the circumference of a circle with center $O$ such that $\angle AOB = 100^o$. If $X$ is the midpoint of minor arc $AB$ and $Y$ is on the circumference of the circle such that $XY\perp AO$, find the measure of $\angle OBY$ . [b]p8. [/b]When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in $6$ hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in $4$ hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates? [b][b]p9.[/b][/b] How many positive integer divisors $n$ of $20000$ are there such that when $20000$ is divided by $n$, the quotient is divisible by a square number greater than $ 1$? [b]p10.[/b] What’s the maximum number of Friday the $13$th’s that can occur in a year? [b]p11.[/b] Let circle $\omega$ pass through points $B$ and $C$ of triangle $ABC$. Suppose $\omega$ intersects segment $AB$ at a point $D \ne B$ and intersects segment $AC$ at a point $E \ne C$. If $AD = DC = 12$, $DB = 3$, and $EC = 8$, determine the length of $EB$. [b]p12.[/b] Let $a,b$ be integers that satisfy the equation $2a^2 - b^2 + ab = 18$. Find the ordered pair $(a,b)$. [b]p13.[/b] Let $a,b,c$ be nonzero complex numbers such that $a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.$ Find $abc -\frac{1}{abc}$ . [b]p14.[/b] Let $\vartriangle ABC$ be an equilateral triangle of side length $1$. Let $\omega_0$ be the incircle of $\vartriangle ABC$, and for $n > 0$, define the infinite progression of circles $\omega_n$ as follows: $\bullet$ $\omega_n$ is tangent to $AB$ and $AC$ and externally tangent to $\omega_{n-1}$. $\bullet$ The area of $\omega_n$ is strictly less than the area of $\omega_{n-1}$. Determine the total area enclosed by all $\omega_i$ for $i \ge 0$. [b]p15.[/b] Determine the remainder when $13^{2020} +11^{2020}$ is divided by $144$. [b]p16.[/b] Let $x$ be a solution to $x +\frac{1}{x}= 1$. Compute $x^{2019} +\frac{1}{x^{2019}}$ . [b]p17. [/b]The positive integers are colored black and white such that if $n$ is one color, then $2n$ is the other color. If all of the odd numbers are colored black, then how many numbers between $100$ and $200$ inclusive are colored white? [b]p18.[/b] What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once? [b]p19.[/b] Let $\vartriangle ABC$ have side lengths $AB = 19$, $BC = 2019$, and $AC = 2020$. Let $D,E$ be the feet of the angle bisectors drawn from $A$ and $B$, and let $X,Y$ to be the feet of the altitudes from $C$ to $AD$ and $C$ to $BE$, respectively. Determine the length of $XY$ . [b]p20.[/b] Suppose I have $5$ unit cubes of cheese that I want to divide evenly amongst $3$ hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese? PS. You had better use hide for answers.