Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO, BO$, and $CO$ intersect the sides of the triangle in the points $A_1, B_1$, and $C_1$. Show that $OA_1 + OB_1 + OC_1 < a$.

During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side? [img]https://cdn.artofproblemsolving.com/attachments/d/2/f8e3a7afb606dfd6fad277f547b116566a4a91.png[/img]

Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions: [list] [*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$. [/list] Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$? [i]Proposed by András Imolay, Budapest[/i]

Let $ABC$ be a triangle inscribed in circle $(O)$ such that points $B, C$ are fixed, while $A$ moves on major arc $BC$ of $(O)$. The tangents through $B$ and $C$ to $(O)$ intersect at $P$. The circle with diameter $OP$ intersects $AC$ and $AB$ at $D$ and $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.

Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$. [i]Proposed by Eugene Chen[/i]

[b]8.1[/b] Find all primes $p,q$ and $r$ such that $$pqr= 5(p + q + r).$$ [b]8.2 [/b] Prove that if $\overline{ab}/\overline{bc} = a/c$, then $$\overline{abb...bb}/\overline{bb...bbc} = a/c$$ (each number has $n$ digits). [b]8.3 / 9.1[/b] Construct a triangle with perimeter, altitude and angle at the base. [b]8.4. / 9.4[/b] Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of $N $squares of non-zero integers. [b]8.5.[/b] In the quadrilateral $ABCD$ the diagonals $AC$ and $BD$ are drawn. Prove that if the circles inscribed in $ABC$ and $ ADC$ touch each other each other, then the circles inscribed in $BAD$ and in $BCD$ also touch each other. [b]8.6 / 9.6[/b] If the numbers $A$ and $n$ are coprime, then there are integers $X$ and $Y$ such that $ |X| <\sqrt{n}$, $|Y| <\sqrt{n} $ and $AX-Y$ is divided by $n$. Prove it. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Kelvin the Nanofrog is visiting his friend, Alex the Nanokat, who lives 483 nanometers away. On his trip to Alex's home, Kelvin travels at $k$ nanometers an hour, where $k$ is an integer, and completes the trip in an integer number of minutes. On his return journey, he travels slower by $7$ nanometers an hour, and completes the trip in an integer number of minutes. What is the smallest total number of minutes Kelvin could have spent traveling?

Use $[x]$ to represent the greatest integer no more than a real number $x$. Let $$S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,)$$ Prove that there are infinitely many $n$ such that $C_n^{S_n}$ is an even number. [b]PS.[/b] [i]Attached is the original wording which forgets left [/i] [b][ [/b][i]. I hope it is ok where I put it.[/i]

Let $n$ be an integer greater than $1$.Find all pairs of integers $(s,t)$ such that equations: $x^n+sx=2007$ and $x^n+tx=2008$ have at least one common real root.

Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.) Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular. [i]Netherlands (Merlijn Staps)[/i]

In a circle $O$, there are six points, $ A$, $ B$, $C$, $D$, $E$, $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$, $BE$, $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE \parallel MN$.

Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.

A semicircle with center $O$ and diameter $AB$ is given. Point $M$ on the extension of $AB$ is taken so that $AM > BM$. A line through $M$ intersects the semicircle at $C$ and $D$ so that $CM < DM$. The circumcircles of triangles $AOD$ and $OBC$ meet again at point $K$. Prove that $OK$ and $KM$ are perpendicular

Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation. For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.

If $i^2=-1$, then $(i-i^{-1})^{-1}=$ $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ -2i \qquad\textbf{(C)}\ 2i \qquad\textbf{(D)}\ -\frac{i}{2} \qquad\textbf{(E)}\ \frac{i}{2}$

In $\vartriangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\vartriangle ABC$ so that each $n$-gon shares a side with a distinct side of $\vartriangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?