Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares.

Let $a$ be an odd digit and $b$ an even digit. Prove that for every positive integer $n$ there exists a positive integer, divisible by $2^n$, whose decimal representation contains no digits other than $a$ and $b$.

The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?

A sequence of positive integers $a_0, a_1, a_2, \dots$ satisfies $a_0 = 83$ and $a_n = (a_{n-1})^{a_{n-1}}$ for all positive integers $n$. Compute the remainder when $a_{2021}$ is divided by $60$.

There are relatively prime positive integers $m$ and $n$ so that \[\frac{m}{n}=\log_4\left(32^{\log_927}\right).\] Find $m+n$.

Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

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 $(n+1)^4 + (n+1)^2 +1$.

A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] \equal{} 6, [\pi] \equal{} 3, [\minus{}1.5] \equal{} \minus{}2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 \plus{} [y]^2 \equal{} 4$.

In a triangle $ABC$, $D$ is the orthogonal projection of the incenter $I$ onto $BC$. Line $DI$ meets the incircle again at $E$. Line $AE$ intersects side $BC$ at point $F$. Suppose that the segment IO is parallel to $BC$, where $O$ is the circumcenter of $\triangle ABC$. If $R$ is the circumradius and $r$ the inradius of the triangle, prove that $EF=2(R-2r)$.

Find all continuous functions $f:[1,8]\to\mathbb R$, such that $$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$

[b]7.1[/b] Construct a trapezoid given four sides. [b]7.2[/b] Prove that $$(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .$$ [b]7.3 [/b] In a quadrilateral $ABCD$, $M$ is the midpoint of AB, $N$ is the midpoint of $CD$. Lines $AD$ and BC intersect $MN$ at points $P$ and $Q$, respectively. Prove that if $\angle BQM = \angle APM$ , then $BC=AD$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png[/img] [b]7.4 / 6.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]7.5 / 8.4[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png[/img] [b]7.6 / 6.5 [/b]The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$. Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence \[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\] is not periodic.

Show tha value $$A=\frac{(b-c)^2}{(a-b)(a-c)}+\frac{(c-a)^2}{(b-c)(b-a)}+\frac{(a-b)^2}{(c-a)(c-b)}$$ does not depend on values of $a$, $b$ and $c$

Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let \[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1. \] Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that: \[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2 \]

On a board $4\times 4$ the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andrés chooses the biggest of each row and Pablo, the biggest of each column. The same number can be chosen by both. Then they are removed from the board all chosen numbers. What is the greatest value that the sum of the numbers can have what are left on the board?

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$

A triple $(x,y,z)$ of real numbers is called a [i]superparticular[/i] if $$\frac{x+1}{x} \cdot \frac{y+1}{y}=\frac{z+1}{z}$$ Find all superparticular positive integer triples.

Let $M$ be the circumcenter of a triangle $ABC.$ The line through $M$ perpendicular to $CM$ meets the lines $CA$ and $CB$ at $Q$ and $P,$ respectively. Prove that \[\frac{\overline{CP}}{\overline{CM}} \cdot \frac{\overline{CQ}}{\overline{CM}}\cdot \frac{\overline{AB}}{\overline{PQ}}= 2.\]

What is the value of $[\log_{10}(5\log_{10}100)]^2$? ${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $

Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps. [i]Proposed by Yannick Yao[/i]

Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]