Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number

Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.

Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.

Let $S$ be a set of positive integers satisfying the following two conditions: • For each positive integer $n$, at least one of $n, 2n, \dots, 100n$ is in $S$. • If $a_1, a_2, b_1, b_2$ are positive integers such that $\gcd(a_1a_2, b_1b_2) = 1$ and $a_1b_1, a_2b_2 \in S,$ then $a_2b_1, a_1b_2 \in S.$ Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\lfloor 10^5r\rfloor$. Note: $S$ has natural density $r$ if $\tfrac{1}{n}|S \cap {1, \dots, n}|$ approaches $r$ as $n$ approaches $\infty$.

Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left? [i]N. Agakhanov[/i]

A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.

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 odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0\equal{}a$ and $ a_n\equal{}a_{n\minus{}1}\plus{}40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely many numbers divisible by $ 2009$.

By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions $ f_0(x),f_1(x),f_2(x),\dots$ such that every number theoretical function $ F(x)$ can be uniquely represented in the form $ F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x)$, $ a_0,a_1,a_2,\dots$ being integers?

We will say that a list of positive integers is [i]admissible [/i] if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that each admissible list can be crossed out some numbers such that the sum of the numbers left uncrossed out is greater than or equal to $1810-d$ and less than or equal to $1810+d$ .

For each positive integer $k$, let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$. Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*} f(d(n+1)) &= d(f(n)+1)\quad \text{and} \\ f(\sigma(n+1)) &= \sigma(f(n)+1) \end{align*} for all positive integers $n$.

[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array. [img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img] [b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$. [img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img] [b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img] [b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ . [b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]1.1.[/b] A circle has a circumference of $20\pi$ inches. Find its area in terms of $\pi$. [b]1.2.[/b] Let $x, y$ be the solution to the system of equations: $x^2 + y^2 = 10 \,\,\, , \,\,\, x = 3y$. Find $x + y$ where both $x$ and $y$ are greater than zero. [b]1. 3.[/b] Chris deposits $\$ 100$ in a bank account. He then spends $30\%$ of the money in the account on biology books. The next week, he earns some money and the amount of money he has in his account increases by $30 \%$. What percent of his original money does he now have? [u]Round 2[/u] [b]2.1.[/b] The bell rings every $45$ minutes. If the bell rings right before the first class and right after the last class, how many hours are there in a school day with $9$ bells? [b]2.2.[/b] The middle school math team has $9$ members. They want to send $2$ teams to ABMC this year: one full team containing 6 members and one half team containing the other $3$ members. In how many ways can they choose a $6$ person team and a $3$ person team? [b]2.3.[/b] Find the sum: $$1 + (1 - 1)(1^2 + 1 + 1) + (2 - 1)(2^2 + 2 + 1) + (3 - 1)(3^2 + 3 + 1) + ...· + (8 - 1)(8^2 + 8 + 1) + (9 - 1)(9^2 + 9 + 1).$$ [u]Round 3[/u] [b]3.1.[/b] In square $ABHI$, another square $BIEF$ is constructed with diagonal $BI$ (of $ABHI$) as its side. What is the ratio of the area of $BIEF$ to the area of $ABHI$? [b]3.2.[/b] How many ordered pairs of positive integers $(a, b)$ are there such that $a$ and $b$ are both less than $5$, and the value of $ab + 1$ is prime? Recall that, for example, $(2, 3)$ and $(3, 2)$ are considered different ordered pairs. [b]3.3.[/b] Kate Lin drops her right circular ice cream cone with a height of $ 12$ inches and a radius of $5$ inches onto the ground. The cone lands on its side (along the slant height). Determine the distance between the highest point on the cone to the ground. [u]Round 4[/u] [b]4.1.[/b] In a Museum of Fine Mathematics, four sculptures of Euler, Euclid, Fermat, and Allen, one for each statue, are nailed to the ground in a circle. Bob would like to fully paint each statue a single color such that no two adjacent statues are blue. If Bob only has only red and blue paint, in how many ways can he paint the four statues? [b]4.2.[/b] Geo has two circles, one of radius 3 inches and the other of radius $18$ inches, whose centers are $25$ inches apart. Let $A$ be a point on the circle of radius 3 inches, and B be a point on the circle of radius $18$ inches. If segment $\overline{AB}$ is a tangent to both circles that does not intersect the line connecting their centers, find the length of $\overline{AB}$. [b]4.3.[/b] Find the units digit to $2017^{2017!}$. [u]Round 5[/u] [b]5.1.[/b] Given equilateral triangle $\gamma_1$ with vertices $A, B, C$, construct square $ABDE$ such that it does not overlap with $\gamma_1$ (meaning one cannot find a point in common within both of the figures). Similarly, construct square $ACFG$ that does not overlap with $\gamma_1$ and square $CBHI$ that does not overlap with $\gamma_1$. Lines $DE$, $FG$, and $HI$ form an equilateral triangle $\gamma_2$. Find the ratio of the area of $\gamma_2$ to $\gamma_1$ as a fraction. [b]5.2.[/b] A decimal that terminates, like $1/2 = 0.5$ has a repeating block of $0$. A number like $1/3 = 0.\overline{3}$ has a repeating block of length $ 1$ since the fraction bar is only over $ 1$ digit. Similarly, the numbers $0.0\overline{3}$ and $0.6\overline{5}$ have repeating blocks of length $ 1$. Find the number of positive integers $n$ less than $100$ such that $1/n$ has a repeating block of length $ 1$. [b]5.3.[/b] For how many positive integers $n$ between $1$ and $2017$ is the fraction $\frac{n + 6}{2n + 6}$ irreducible? (Irreducibility implies that the greatest common factor of the numerator and the denominator is $1$.) [u]Round 6[/u] [b]6.1.[/b] Consider the binary representations of $2017$, $2017 \cdot 2$, $2017 \cdot 2^2$, $2017 \cdot 2^3$, $... $, $2017 \cdot 2^{100}$. If we take a random digit from any of these binary representations, what is the probability that this digit is a $1$ ? [b]6.2.[/b] Aaron is throwing balls at Carlson’s face. These balls are infinitely small and hit Carlson’s face at only $1$ point. Carlson has a flat, circular face with a radius of $5$ inches. Carlson’s mouth is a circle of radius $ 1$ inch and is concentric with his face. The probability of a ball hitting any point on Carlson’s face is directly proportional to its distance from the center of Carlson’s face (so when you are $2$ times farther away from the center, the probability of hitting that point is $2$ times as large). If Aaron throws one ball, and it is guaranteed to hit Carlson’s face, what is the probability that it lands in Carlson’s mouth? [b]6.3.[/b] The birth years of Atharva, his father, and his paternal grandfather form a geometric sequence. The birth years of Atharva’s sister, their mother, and their grandfather (the same grandfather) form an arithmetic sequence. If Atharva’s sister is $5$ years younger than Atharva and all $5$ people were born less than $200$ years ago (from $2017$), what is Atharva’s mother’s birth year? [u]Round 7[/u] [b]7. 1.[/b] A function $f$ is called an “involution” if $f(f(x)) = x$ for all $x$ in the domain of $f$ and the inverse of $f$ exists. Find the total number of involutions $f$ with domain of integers between $ 1$ and $ 8$ inclusive. [b]7.2.[/b] The function $f(x) = x^3$ is an odd function since each point on $f(x)$ corresponds (through a reflection through the origin) to a point on $f(x)$. For example the point $(-2, -8)$ corresponds to $(2, 8)$. The function $g(x) = x^3 - 3x^2 + 6x - 10$ is a “semi-odd” function, since there is a point $(a, b)$ on the function such that each point on $g(x)$ corresponds to a point on $g(x)$ via a reflection over $(a, b)$. Find $(a, b)$. [b]7.3.[/b] A permutations of the numbers $1, 2, 3, 4, 5$ is an arrangement of the numbers. For example, $12345$ is one arrangement, and $32541$ is another arrangement. Another way to look at permutations is to see each permutation as a function from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$. For example, the permutation $23154$ corresponds to the function f with $f(1) = 2$, $f(2) = 3$, $f(3) = 1$, $f(5) = 4$, and $f(4) = 5$, where $f(x)$ is the $x$-th number of the permutation. But the permutation $23154$ has a cycle of length three since $f(1) = 2$, $f(2) = 3$, $f(3) = 1$, and cycles after $3$ applications of $f$ when regarding a set of $3$ distinct numbers in the domain and range. Similarly the permutation $32541$ has a cycle of length three since $f(5) = 1$, $f(1) = 3$, and $f(3) = 5$. In a permutation of the natural numbers between $ 1$ and $2017$ inclusive, find the expected number of cycles of length $3$. [u]Round 8[/u] [b]8.[/b] Find the number of characters in the problems on the accuracy round test. This does not include spaces and problem numbers (or the periods after problem numbers). For example, “$1$. What’s $5 + 10$?” would contain $11$ characters, namely “$W$,” “$h$,” “$a$,” “$t$,” “$’$,” “$s$,” “$5$,” “$+$,” “$1$,” “$0$,” “?”. If the correct answer is $c$ and your answer is $x$, then your score will be $$\max \left\{ 0, 13 -\left\lceil \frac{|x-c|}{100} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$, all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$. for all positive integers $n>1$, prove that there exists a good set $S$ such that $n \not \in S$. Proposed by Seyed Reza Hosseini Dolatabadi

a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$). b) Let $n$ be a natural number. Find the number of [i]square free[/i] numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is [i]square free[/i] if it's not divisible by any square of a prime number).

Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a$, $b$, and $c$ is $2008$).

Find all $n$ such that \[n = d (n) ^ 4\] Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2$.

Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$ \[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]

Let $\tau(n)$ denote the number of positive integer divisors of a positive integer $n$ (for example, $\tau(2022) = 8$). Given a polynomial $P(X)$ with integer coefficients, we define a sequence $a_1, a_2,\ldots$ of nonnegative integers by setting \[a_n =\begin{cases}\gcd(P(n), \tau (P(n)))&\text{if }P(n) > 0\\0 &\text{if }P(n) \leq0\end{cases}\] for each positive integer $n$. We then say the sequence [i]has limit infinity[/i] if every integer occurs in this sequence only finitely many times (possibly not at all). Does there exist a choice of $P(X)$ for which the sequence $a_1$, $a_2$, . . . has limit infinity? [i]Jovan Vuković[/i]

Let $\mathbb N$ denote the set of positive integers. A function $f\colon\mathbb N\to\mathbb N$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers \[f(m+1),f(m+2),\ldots,f(m+f(n))\] is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$.

Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$.