Prove that the equation $x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y$ has no solution in the set of the positive integers.

Let $p$ be an odd prime number. Prove that there exist nonnegative integers $x,y,z,t$ not all of which are $0$ such that $t<p$ and \[x^2+y^2+z^2=tp.\]

Prove that for each positive integer $a$ there exists such an integer $b>a$, for which $1+2^a+3^a$ divides $1+2^b+3^b$.

Let $m_{1},m_{2},\ldots,m_{k}$ be integers with $2\leq m_{1}$ and $2m_{1}\leq m_{i+1}$ for all $i$. Show that for any integers $a_{1},a_{2},\ldots,a_{k}$ there are infinitely many integers $x$ which do not satisfy any of the congruences \[x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.\]

A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.

Two different prime numbers $p$ and $q$ differ in less than $2$ times. Prove that exists two consecutive natural numbers, such that largest prime divisor of first number is $p$, and largest prime divisor of second number is $q$.

Find the largest possible sum $ m + n$ for positive integers $m, n \le 100$ such that $m + 1 \equiv 3$ (mod $4$) and there exists a prime number $p$ and nonnegative integer $a$ such $\frac{m^{2n-1}-1}{m-1} = m^n+p^a$ .

Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that: \[d_6^{2} +d_7^{2} - 1=n\]

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$

Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.

Find all prime numbers $p, q, r$ such that \[15p+7pq+qr=pqr.\]

[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccccc} & & & a & b & c & d \\ x & & & & & a & b \\ \hline & & c & d & b & d & b \\ + & c & e & b & f & b & \\ \hline & c & g & a & e & g & b \\ \end{tabular}$ [b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$? [b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$ [b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

For every positive integer $N$, let $\sigma(N)$ denote the sum of the positive integer divisors of $N$. Find all integers $m\geq n\geq 2$ satisfying \[\frac{\sigma(m)-1}{m-1}=\frac{\sigma(n)-1}{n-1}=\frac{\sigma(mn)-1}{mn-1}.\] [i]Ankan Bhattacharya[/i]

[b]p1.[/b] A robot is at position $0$ on a number line. Each second, it randomly moves either one unit in the positive direction or one unit in the negative direction, with probability $\frac12$ of doing each. Find the probability that after $4$ seconds, the robot has returned to position $0$. [b]p2.[/b] How many positive integers $n \le 20$ are such that the greatest common divisor of $n$ and $20$ is a prime number? [b]p3.[/b] A sequence of points $A_1$, $A_2$, $A_3$, $...$, $A_7$ is shown in the diagram below, with $A_1A_2$ parallel to $A_6A_7$. We have $\angle A_2A_3A_4 = 113^o$, $\angle A_3A_4A_5 = 100^o$, and $\angle A_4A_5A_6 = 122^o$. Find the degree measure of $\angle A_1A_2A_3 + \angle A_5A_6A_7$. [center][img]https://cdn.artofproblemsolving.com/attachments/d/a/75b06a6663b2f4258e35ef0f68fcfbfaa903f7.png[/img][/center] [b]p4.[/b] Compute $$\log_3 \left( \frac{\log_3 3^{3^{3^3}}}{\log_{3^3} 3^{3^3}} \right)$$ [b]p5.[/b] In an $8\times 8$ chessboard, a pawn has been placed on the third column and fourth row, and all the other squares are empty. It is possible to place nine rooks on this board such that no two rooks attack each other. How many ways can this be done? (Recall that a rook can attack any square in its row or column provided all the squares in between are empty.) [b]p6.[/b] Suppose that $a, b$ are positive real numbers with $a > b$ and $ab = 8$. Find the minimum value of $\frac{a^2+b^2}{a-b} $. [b]p7.[/b] A cone of radius $4$ and height $7$ has $A$ as its apex and $B$ as the center of its base. A second cone of radius $3$ and height $7$ has $B$ as its apex and $A$ as the center of its base. What is the volume of the region contained in both cones? [b]p8.[/b] Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$ be a permutation of the numbers $1$, $2$, $3$, $4$, $5$, $6$. We say $a_i$ is visible if $a_i$ is greater than any number that comes before it; that is, $a_j < a_i$ for all $j < i$. For example, the permutation $2$, $4$, $1$, $3$, $6$, $5$ has three visible elements: $2$, $4$, $6$. How many such permutations have exactly two visible elements? [b]p9.[/b] Let $f(x) = x+2x^2 +3x^3 +4x^4 +5x^5 +6x^6$, and let $S = [f(6)]^5 +[f(10)]^3 +[f(15)]^2$. Compute the remainder when $S$ is divided by $30$. [b]p10.[/b] In triangle $ABC$, the angle bisector from $A$ and the perpendicular bisector of $BC$ meet at point $D$, the angle bisector from $B$ and the perpendicular bisector of $AC$ meet at point $E$, and the perpendicular bisectors of $BC$ and $AC$ meet at point $F$. Given that $\angle ADF = 5^o$, $\angle BEF = 10^o$, and $AC = 3$, find the length of $DF$. [img]https://cdn.artofproblemsolving.com/attachments/6/d/6bb8409678a4c44135d393b9b942f8defb198e.png[/img] [b]p11.[/b] Let $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$. How many subsets $S$ of $\{1, 2,..., 2011\}$ are there such that $$F_{2012} - 1 =\sum_{i \in S}F_i?$$ [b]p12.[/b] Let $a_k$ be the number of perfect squares $m$ such that $k^3 \le m < (k + 1)^3$. For example, $a_2 = 3$ since three squares $m$ satisfy $2^3 \le m < 3^3$, namely $9$, $16$, and $25$. Compute$$ \sum^{99}_{k=0} \lfloor \sqrt{k}\rfloor a_k, $$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p13.[/b] Suppose that $a, b, c, d, e, f$ are real numbers such that $$a + b + c + d + e + f = 0,$$ $$a + 2b + 3c + 4d + 2e + 2f = 0,$$ $$a + 3b + 6c + 9d + 4e + 6f = 0,$$ $$a + 4b + 10c + 16d + 8e + 24f = 0,$$ $$a + 5b + 15c + 25d + 16e + 120f = 42.$$ Compute $a + 6b + 21c + 36d + 32e + 720f.$ [b]p14.[/b] In Cartesian space, three spheres centered at $(-2, 5, 4)$, $(2, 1, 4)$, and $(4, 7, 5)$ are all tangent to the $xy$-plane. The $xy$-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation $ax + by + cz = d$ for some real numbers $a$, $b$, $c$, $d$. Find $\frac{c}{a}$ . [b]p15.[/b] Find the number of pairs of positive integers $a$, $b$, with $a \le 125$ and $b \le 100$, such that $a^b - 1$ is divisible by $125$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Call an ordered pair $(a, b)$ of positive integers [i]fantastic [/i] if and only if $a, b \le 10^4$ and $$gcd(a \cdot n! - 1, a \cdot (n + 1)! + b) > 1$$ for infinitely many positive integers $n$. Find the sum of $a + b$ across all fantastic pairs $(a, b)$.

A positive integer $n \ge 2$ is called [i]peculiar [/i] if the number $n \choose i$ + $n \choose j $ $-i-j$ is even for all integers $i$ and $j$ such that $0 \le i \le j \le n$. Determine all peculiar numbers.

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

[u]Set 7[/u] [b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square. [b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees. [b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square. PS. You should use hide for answers.

Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$. [i]A. Golovanov[/i]

For any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\sum^{\lfloor \log_{23} n\rfloor}_{i=1} s_{20} \left( \left\lfloor \frac{n}{23^i} \right\rfloor \right)= 103 \,\,\, \text{and} \,\,\, \sum^{\lfloor \log_{20} n\rfloor}_{i=1} s_{23} \left( \left\lfloor \frac{n}{20^i} \right\rfloor \right)= 115$$ Compute $s_{20}(n) - s_{23}(n)$.

Let $r \geq 2023$ be a rational number. The real numbers $a, b$, and $c$ satisfy \[ 4a^2 + 4b^2 + 9c^2 = r. \] Does there exist a value of $r$ for which the number of rational triples $(a,b,c)$ that achieve the maximum possible value of $4ab + 6bc - 6ac$ is: a) zero b) finite, but non-zero?

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 $n$ ).

a) Is it possible to arrange numbers $1,2,...,13$ in a circumference such that the sum of any two neighbouring numbers to be a prime number? b) Is the same problem possible for the numbers $1,2,...,16$?