$4.$ Let $N$ be an integer greater than $1$.Denote by $x$ the smallest positive integer with the following property:there exists a positive integer $y$ strictly less than $x-1$ , such that $x$ divides $N+y$.Prove that x is either $p^n$ or $2p$ , where $p$ is a prime number and $n$ is a positive integer

Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.

The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?

Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.

Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$. [i]Proposed by Mark Sellke[/i]

[u]Set 4[/u] [b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$. [b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$. [b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$. [u]Set 5[/u] [b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$. [b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$. [b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 6[/u] [b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$. [b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$? [b]G18.[/b] Deja vu? Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.

Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

Prove that for any natural number $ n $ there are $ n $ consecutive numbers, each one of these numbers having the following property: the sum of the positive divisors of a number $ x $ is greater than $ 2x. $

[u]Round 1[/u] [b]p1.[/b] When Shiqiao sells a bale of kale, he makes $x$ dollars, where $$x =\frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8}{3 + 4 + 5 + 6}.$$ Find $x$. [b]p2.[/b] The fraction of Shiqiao’s kale that has gone rotten is equal to $$\sqrt{ \frac{100^2}{99^2} -\frac{100}{99}}.$$ Find the fraction of Shiqiao’s kale that has gone rotten. [b]p3.[/b] Shiqiao is growing kale. Each day the number of kale plants doubles, but $4$ of his kale plants die afterwards. He starts with $6$ kale plants. Find the number of kale plants Shiqiao has after five days. [u]Round 2[/u] [b]p4.[/b] Today the high is $68$ degrees Fahrenheit. If $C$ is the temperature in Celsius, the temperature in Fahrenheit is equal to $1.8C + 32$. Find the high today in Celsius. [b]p5.[/b] The internal angles in Evan’s triangle are all at most $68$ degrees. Find the minimum number of degrees an angle of Evan’s triangle could measure. [b]p6.[/b] Evan’s room is at $68$ degrees Fahrenheit. His thermostat has two buttons, one to increase the temperature by one degree, and one to decrease the temperature by one degree. Find the number of combinations of $10$ button presses Evan can make so that the temperature of his room never drops below $67$ degrees or rises above $69$ degrees. [u]Round 3[/u] [b]p7.[/b] In a digital version of the SAT, there are four spaces provided for either a digit $(0-9)$, a fraction sign $(\/)$, or a decimal point $(.)$. The answer must be in simplest form and at most one space can be a non-digit character. Determine the largest fraction which, when expressed in its simplest form, fits within this space, but whose exact decimal representation does not. [b]p8.[/b] Rounding Rox picks a real number $x$. When she rounds x to the nearest hundred, its value increases by $2.71828$. If she had instead rounded $x$ to the nearest hundredth, its value would have decreased by $y$. Find $y$. [b]p9.[/b] Let $a$ and $b$ be real numbers satisfying the system of equations $$\begin{cases} a + \lfloor b \rfloor = 2.14 \\ \lfloor a \rfloor + b = 2.72 \end{cases}$$ Determine $a + b$. [u]Round 4[/u] [b]p10.[/b] Carol and Lily are playing a game with two unfair coins, both of which have a $1/4$ chance of landing on heads. They flip both coins. If they both land on heads, Lily loses the game, and if they both land on tails, Carol loses the game. If they land on different sides, Carol and Lily flip the coins again. They repeat this until someone loses the game. Find the probability that Lily loses the game. [b]p11.[/b] Dongchen is carving a circular coin design. He carves a regular pentagon of side length $1$ such that all five vertices of the pentagon are on the rim of the coin. He then carves a circle inside the pentagon so that the circle is tangent to all five sides of the pentagon. Find the area of the region between the smaller circle and the rim of the coin. [b]p12.[/b] Anthony flips a fair coin six times. Find the probability that at some point he flips $2$ heads in a row. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3248731p29808147]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square.

Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.

Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.

Prove that there exist infinitely many positive integers which cannot be presented in the form $x_1^3+x_2^5+x_3^7+x_4^9+x_5^{11}$ where $x_1,x_2,x_3,x_4,x_5$ are positive integers. (V. Bernik)

[b]8.1[/b] In the parallelogram $ABCD$ , the diagonal $AC$ is greater than the diagonal $BD$. The point $M$ on the diagonal $AC$ is such that around the quadrilateral $BCDM$ one can circumscribe a circle. Prove that $BD$ is the common tangent of the circles circumscribed around the triangles $ABM$ and $ADM$. [img]https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png[/img] [b]8.2 [/b] $A$ is an odd integer, $x$ and $y$ are roots of equation $t^2+At-1=0$. Prove that $x^4 + y^4$ and $x^5+ y^5$ are coprime integer numbers. [b]8.3[/b] A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made. [b]8.4 /7.6[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. [b]8.5 [/b] All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way. [url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6*[/url] (asterisk problems in separate posts) PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$. Malik Talbi

The sequence $a_1, a_2, ...$ of natural numbers satisfies $GCD(a_i, a_j)=GCD(i, j)$ for all $i \neq j$. Prove that $a_i=i$ for all $i$.

Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$ Find the numerical value of $a + b + c$.

Find all primes $p, q$ such that $p + q = (p-q)^3$.

Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.

This sequence lists the perfect squares in increasing order: \[0,1,4,9,16,\cdots ,a,10^8,b,\cdots\] Determine the value of $b-a$.

Prove that there exist infinitely many natural numbers $ n$ such that the numerator of $ 1 \plus{} \frac {1}{2} \plus{} \frac {1}{3} \plus{} \frac {1}{4} \plus{} ... \plus{} \frac {1}{n}$ in the lowest terms is not a power of a prime number.

Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.