Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?

We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$ a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$. b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$. c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$. d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$. You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

The sequence $(p_n)_{n\in\mathbb N}$ is defined by $p_1=2$ and, for $n\ge2$, $p_n$ is the largest prime factor of $p_1p_2\cdots p_{n-1}+1$. Show that $p_n\ne5$ for all $n$.

Let $f(x)$ be a function acting on a string of $0$s and $1$s, defined to be the number of substrings of $x$ that have at least one $1$, where a substring is a contiguous sequence of characters in $x$. Let $S$ be the set of binary strings with $24$ ones and $100$ total digits. Compute the maximum possible value of $f(s)$ over all $s\in S$. For example, $f(110) = 5$ as $\underline{1}10$, $1\underline{1}0$, $\underline{11}0$, $1\underline{10}$, and $\underline{110}$ are all substrings including a $1$. Note that $11\underline{0}$ is not such a substring.

Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.

Let $p$ be a prime number, and let $n$ be a positive integer. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i\in \{0, 1, \ldots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that \[p^n \mid (a_1a_2 + a_3a_4 + 1).\]

Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\frac{1}{z}=5,$ and $y+\frac{1}{x}=29.$ Then $z+\frac{1}{y}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

For a finite set of naturals $(C)$, the product of its elements is going to be noted $P(C)$. We are going to define $P (\phi) = 1$. Calculate the value of the expression $$\sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)}$$

Is there a natural number $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change.

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]

(a) Prove that the set $X = (1,2,....100)$ cannot be partitoned into THREE subsets such that two numbers differing by a square belong to different subsets. (b) Prove that $X$ can so be partitioned into $5$ subsets.

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

Find all primes $p$ and positive integers $n$ satisfying \[n \cdot 5^{n-n/p} = p! (p^2+1) + n.\]

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.

Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$

Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

Find all positive integers $(a, b)$, such that $\frac{a^2}{2ab^2-b^3+1}$ is also a positive integer.

Does there exist a real number $r$ such that the equation $$x^3-2023x^2-2023x+r=0$$ has three distinct rational roots?

Find the maximum value of natural components of number $96$ that we can seperate such that all of them must be relatively prime number withh each other.

Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

Prove that there exist an infinite number of triples $n-1 $,$n$,$n + 1$ such that (a) $n$ can be represented as the sum of two squares of natural numbers but neither of $n-1$ and $n+1$ can; (b) each of these three numbers can be represented as the sum of two squares. (V Senderov)