Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$. a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \le k \le 1992, $ but $T(1993 \cdot N)$ is odd. b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.

Koshchey opened an account at the bank. Initially, it had 0 rubles. On the first day, Koshchey puts $k>0$ rubles in, and every next day adds one ruble more there than the day before. Each time after Koshchey deposits money into the account, the total amount in the account is divided by two by the bank. Find all such $k{}$ for which the amount on the account will always be an integer number of rubles. [i]Proposed by S. Berlov[/i]

Let $k$ and $n$ be positive integers with $n>2$. Show that the equation: \[x^n-y^n=2^k\] has no positive integer solutions.

Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

Prove that the set of integers of the form $2^{k}-3$ ($k=2,3,\cdots$) contains an infinite subset in which every two members are relatively prime.

Let $n$ be a positive integer and $M = {1, 2, . . . , 2n + 1}$. Find out in how many ways we can split the set $M$ into three mutually disjoint nonempty sets $A,B,C$ so that both the following are true: (i) for each $a \in A$ and $b \in B$, the remainder of the division of $a$ by $b$ belongs to $C$ (ii) for each $c \in C$ there exists $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.

The numbers $1,2,3,\dots,1000$ are written on the board. Patya and Vassya are playing a game. They take turn alternatively erasing a number from the board. Patya begins. If after a turn all numbers (maybe one) on the board be divisible by a natural number greater than $1$ the player who last played loses. If after some number of steps the only remaining number on the board be $1$ then they call it a draw. Determine the result of the game if they both play their best.

Let $S(k)$ denote the digit-sum of a positive integer $k$(in base $10$). Determine the smallest positive integer $n$ such that \[S(n^2 ) = S(n) - 7\]

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

[u]Round 9[/u] [b]p25. [/b]Define a sequence $\{a_n\}_{n \ge 1}$ of positive real numbers by $a_1 = 2$ and $a^2_n -2a_n +5 =4a_{n-1}$ for $n \ge 2$. Suppose $k$ is a positive real number such that $a_n <k$ for all positive integers $n$. Find the minimum possible value of $k$. [b]p26.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $C A = 15$. Suppose the incenter of $\vartriangle ABC$ is $I$ and the incircle is tangent to $BC$ and $AB$ at $D$ and $E$, respectively. Line $\ell$ passes through the midpoints of $BD$ and $BE$ and point $X$ is on $\ell$ such that $AX \parallel BC$. Find $X I$ . [b]p27.[/b] Let $x, y, z$ be positive real numbers such that $x y + yz +zx = 20$ and $x^2yz +x y^2z +x yz^2 = 100$. Additionally, let $s = \max (x y, yz,xz)$ and $m = \min(x, y, z)$. If $s$ is maximal, find $m$. [u]Round 10[/u] [b]p28.[/b] Let $\omega_1$ be a circle with center $O$ and radius $1$ that is internally tangent to a circle $\omega_2$ with radius $2$ at $T$ . Let $R$ be a point on $\omega_1$ and let $N$ be the projection of $R$ onto line $TO$. Suppose that $O$ lies on segment $NT$ and $\frac{RN}{NO} = \frac4 3$ . Additionally, let $S$ be a point on $\omega_2$ such that $T,R,S$ are collinear. Tangents are drawn from $S$ to $\omega_1$ and touch $\omega_1$ at $P$ and $Q$. The tangent to $\omega_1$ at $R$ intersects $PQ$ at $Z$. Find the area of triangle $\vartriangle ZRS$. [b]p29.[/b] Let $m$ and $n$ be positive integers such that $k =\frac{ m^2+n^2}{mn-1}$ is also a positive integer. Find the sum of all possible values of $k$. [b]p30.[/b] Let $f_k (x) = k \cdot \ min (x,1-x)$. Find the maximum value of $k \le 2$ for which the equation $f_k ( f_k ( f_k (x))) = x$ has fewer than $8$ solutions for $x$ with $0 \le x \le 1$. [u]Round 11[/u] In the following problems, $A$ is the answer to Problem $31$, $B$ is the answer to Problem $32$, and $C$ is the answer to Problem $33$. For this set, you should find the values of $A$,$B$, and $C$ and submit them as answers to problems $31$, $32$, and $33$, respectively. Although these answers depend on each other, each problem will be scored separately. [b]p31.[/b] Find $$A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}$$ [b]p32.[/b] Let $D = 7 \cdot B \cdot C$. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of $r$ units clockwise along the circle, where $r$ is picked uniformly at random from the interval $\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]$. Then, the entire unit circle is rotated $\frac{\pi}{4}$ radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the $x$-axis for the first two turns. Find the probability that the ant wins the game. [b]p33.[/b] Let $m$ and $n$ be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer $2016 \cdot B$, respectively. Find $\frac{n^2}{m^2 - mn}$. [u]Round 12[/u] [b]p34.[/b] There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ [b]p35.[/b] Estimate the value of $$\log_{10} \left(\prod_{k|2016} k!\right), $$ where the product is taken over all positive divisors $k$ of $2016$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] Estimate the value of $\sqrt{2016}^{\sqrt[4]{2016}}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$, $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$, satisfy \[ f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1, \] and are zero elsewhere.

Given a positive integer $h$, show that there are a finite number of triangles with integer sides $a, b, c$ and altitude relative to side $c$ equal to $h$ .

Find all positive integers $k > 1$ for which there exists a positive integer $n$ such that $\tbinom{n}{k}$ is divisible by $n$, and $\tbinom{n}{m}$ is not divisible by $n$ for $2\leq m < k$. [i]Merlijn Staps[/i]

Prove that no perfect cube is of the form $y^2+108$ where $y \in \mathbb{Z}$.

Prove that there exists only one pair $ (p,q) $ of odd primes satisfying the properties that $ p^2\equiv 4\pmod q $ and $ q^2\equiv 1\pmod p. $ [i]Ana Maria Acu[/i]

You are initially given the number $n=1$. Each turn, you may choose any positive divisor $d\mid n$, and multiply $n$ by $d+1$. For instance, on the first turn, you must select $d=1$, giving $n=1\cdot(1+1)=2$ as your new value of $n$. On the next turn, you can select either $d=1$ or $2$, giving $n=2\cdot(1+1)=4$ or $n=2\cdot(2+1)=6$, respectively, and so on. Find an algorithm that, in at most $k$ steps, results in $n$ being divisible by the number $2021^{2021^{2021}} - 1$. An algorithm that completes in at most $k$ steps will be awarded: 1 pt for $k>2021^{2021^{2021}}$ 20 pts for $k=2021^{2021^{2021}}$ 50 pts for $k=10^{10^4}$ 75 pts for $k=10^{10}$ 90 pts for $k=10^5$ 95 pts for $k=6\cdot10^4$ 100 pts for $k=5\cdot10^4$

[b]numbers $n^2+1$[/b] Prove that there are infinitely many natural numbers of the form $n^2+1$ such that they don't have any divisor of the form $k^2+1$ except $1$ and themselves. time allowed for this question was 45 minutes.

Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.

There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$. (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.

[b]p1.[/b] What is $20\times 20 - 19\times 19$? [b]p2.[/b] Andover has a total of $1440$ students and teachers as well as a $1 : 5$ teacher-to-student ratio (for every teacher, there are exactly $5$ students). In addition, every student is either a boarding student or a day student, and $70\%$ of the students are boarding students. How many day students does Andover have? [b]p3.[/b] The time is $2:20$. If the acute angle between the hour hand and the minute hand of the clock measures $x$ degrees, find $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png[/img] [b]p4.[/b] Point $P$ is located on segment $AC$ of square $ABCD$ with side length $10$ such that $AP >CP$. If the area of quadrilateral $ABPD$ is $70$, what is the area of $\vartriangle PBD$? [b]p5.[/b] Andrew always sweetens his tea with sugar, and he likes a $1 : 7$ sugar-to-unsweetened tea ratio. One day, he makes a $100$ ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a $1 : 2$ sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness? [b]p6.[/b] Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly $2020$ meters. He wants to raise the entire track $6$ meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of $2$ meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground. [b]p7.[/b] Mr. DoBa writes the numbers $1, 2, 3,..., 20$ on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining $18$ numbers is exactly $11$. What is the maximum possible value of the larger of the two numbers that Will erased? [b]p8.[/b] Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number? [b]p9.[/b] Let $S$ be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, $1$, $24$, and $369$ are all elements of $S$, while $20$ and $667$ are not. If the elements of $S$ are written in increasing order, what is the $100$th number written? [b]p10.[/b] Find the largest prime factor of the expression $2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1$. [b]p11.[/b] Christina writes down all the numbers from $1$ to $2020$, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down? [b]p12.[/b] Triangle $ABC$ has side lengths $AB = AC = 10$ and $BC = 16$. Let $M$ and $N$ be the midpoints of segments $BC$ and $CA$, respectively. There exists a point $P \ne A$ on segment $AM$ such that $2PN = PC$. What is the area of $\vartriangle PBC$? [b]p13.[/b] Consider the polynomial $$P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.$$ Let its four roots be $a, b, c, d$. Evaluate the expression $$(a + b + c)(a + b + d)(a + c + d)(b + c + d).$$ [b]p14.[/b] Consider the system of equations $$|y - 1| = 4 -|x - 1|$$ $$|y| =\sqrt{|k - x|}.$$ Find the largest $k$ for which this system has a solution for real values $x$ and $y$. [b]p16.[/b] Let $T_n = 1 + 2 + ... + n$ denote the $n$th triangular number. Find the number of positive integers $n$ less than $100$ such that $n$ and $T_n$ have the same number of positive integer factors. [b]p17.[/b] Let $ABCD$ be a square, and let $P$ be a point inside it such that $PA = 4$, $PB = 2$, and $PC = 2\sqrt2$. What is the area of $ABCD$? [b]p18.[/b] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_n$ for all integers $n \ge 0$. Let $$ S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... $$ Compute $420S$. [b]p19.[/b] Let $ABCD$ be a square with side length $5$. Point $P$ is located inside the square such that the distances from $P$ to $AB$ and $AD$ are $1$ and $2$ respectively. A point $T$ is selected uniformly at random inside $ABCD$. Let $p$ be the probability that quadrilaterals $APCT$ and $BPDT$ are both not self-intersecting and have areas that add to no more than $10$. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. Note: A quadrilateral is self-intersecting if any two of its edges cross. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that \[ 4n^6 \plus{} n^3 \plus{} 5\] is divisible by $ 7$.

Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)