An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point

[color=darkred]Let $a,b\in\mathbb{R}$ with $0<a<b$ . Prove that: [list] [b]a)[/b] $2\sqrt {ab}\le\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\le a+b$ , for $x,y,z\in [a,b]\, .$ [b]b)[/b] $\left\{\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\, |\, x,y,z\in [a,b]\right\}=[2\sqrt {ab},a+b]\, .$ [/list][/color]

Given three non-collinear points in the plane, find a line which is equally distant from each of the points. How many such lines are there?

Given two positive integers $m$ and $n$, find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ of them who form $m$ pairs of mutually acquainted people or there are $2n$ of them forming $n$ pairs of mutually unacquainted people.

Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_0, P_1, \ldots, P_N$ in the plane. After choosing $P_k$ (but before choosing $P_{k+1}$) for $k \geq 1$, Bob tells "warmer'' if $P_k$ is closer to $B$ than $P_{k-1}$, otherwise he says "colder''. After Alice has chosen $P_N$ and heard Bob's answer, Alice chooses a final point $A$. Alice wins if the distance $AB$ is at most $\frac 1 {2020}$, otherwise Bob wins. Show that if $N=18$, Alice cannot guarantee a win.

Prove that every positive integer which is not a member of the infinite set below is equal to the sum of two or more distinct numbers of the set \[\{ 3,-2, 2^{2}3,-2^{3}, \cdots, 2^{2k}3,-2^{2k+1}, \cdots \}=\{3,-2, 12,-8, 48,-32, 192, \cdots \}.\]

Let be given an open interval $(\alpha; \beta)$ with $\alpha - \beta = \frac{1}{27}$. Determine the maximum number of irreducible fractions $\frac{a}{b}$ in $(\alpha; \beta)$ with $1 \leq b \leq 2007$?

For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to \[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\] [i]Proposed by Daniel Zhu.[/i]

Let $\alpha\in(\frac{\pi}{4},\frac{\pi}{2})$, then the order of $(\cos\alpha)^{\cos\alpha},(\sin\alpha)^{\cos\alpha},(\cos\alpha)^{\sin\alpha}$ is $\text{(A)}(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}$ $\text{(B)}(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}<(\sin\alpha)^{\cos\alpha}$ $\text{(C)}(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}$ $\text{(D)}(\cos\alpha)^{\sin\alpha}<(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}$

Let $n$ be a natural number with $120$ positive divisors (including $1$ and $n$). For each divisor $d$ of $n$, let $q$ be the quotient and $r$ the remainder when dividing $4n - 3$ by $d$. Let $Q$ be the sum of all the quotients $q$, and $R$ the sum of all the remainders $r$ for the $120$ divisions of $4n - 3$ by $d$. Determine all posible values of $Q - 4R$

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $$AB+CD>DA+BC.$$

For all $n\in\mathbb{N}$, we denote by $s(n)$ the sum of its digits. Find all integers $k\geq 2$ such that there exist $a,b\in\mathbb{N}$ with $$s(n^3+an+b)\equiv s(n)\pmod k,$$ for all $n\in\mathbb{N}^*$.

Which statement does not describe benzene, $\ce{C6H6}$? $ \textbf{(A)}\ \text{It is an aromatic hydrocarbon.} \qquad$ $\textbf{(B)}\ \text{It exists in two isomeric forms.} \qquad$ $\textbf{(C)}\ \text{It undergoes substitution reactions.} \qquad$ $\textbf{(D)}\ \text{It can react to form three different products with the formula C}_6\text{H}_4\text{Cl}_2\qquad$

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $ \$4$ per pair and each T-shirt costs $ \$5$ more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $ \$2366$, how many members are in the League? $ \textbf{(A)}\ 77 \qquad \textbf{(B)}\ 91 \qquad \textbf{(C)}\ 143 \qquad \textbf{(D)}\ 182 \qquad \textbf{(E)}\ 286$

Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?

Find a costant $C$, such that $$ \frac{S}{ab+bc+ca}\le C$$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle. (The maximal number of points is given for the best possible constant, with proof.)

[b]p1.[/b] Ivan Ivanovich came to the store with $20$ rubles. The store sold brooms for $1$ ruble. $17$ kopecks and basins for $1$ rub. $66$ kopecks (there are no other products left in the store). How many brooms and how many basins does he need to buy in order to spend as much money as possible? (Note: $1$ ruble = $100$ kopecks) [b]p2.[/b] On the road from city A to city B there are kilometer posts. On each pillar, on one side, the distance to city A is written, and on the other, to B. In the morning, a tourist passed by a pillar on which one number was twice the size of the other. After walking another $10$ km, the tourist saw a post on which the numbers differed exactly three times. What is the distance from A to B? List all possibilities. [b]p3.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365 days on the next New Year's Eve? [b]p4.[/b] What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from $100$ to $999$? [b]p5.[/b] An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to the denominator of the third fraction, the result was $13/23$. What fraction was written on the board? [b]p6.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] The square is divided by straight lines into $25$ rectangles (fig. 1). The areas of some of them are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p9.[/b] Petya multiplied all natural numbers from $1$ to his age inclusive. The result is a number $$8 \,\, 841 \,\,761993 \,\,739 \,\,701954 \,\,543 \,\,616 \,\,000 \,\,000.$$ How old is Petya? [b]p10.[/b] There are $100$ integers written in a line, and the sum of any three in a row is equal to $10$ or $11$. The first number is equal to one. What could the last number be? List all possibilities. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

There are $23$ balls on a table, all of which are either red or blue, such that the probability that there are $n$ red balls and $23-n$ blue balls on the table ($1 \le n \le 22$) is proportional to $n$. (e.g. the probability that there are $2$ red balls and $21$ blue balls is twice the probability that there are $1$ red ball and $22$ blue balls.) Given that the probability that the red balls and blue balls can be arranged in a line such that there is a blue ball on each end, no two red balls are next to each other, and an equal number of blue balls can be placed between each pair of adjacent red balls is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. Note: There can be any nonzero number of consecutive blue balls at the ends of the line. [i]Proposed by Ada Tsui[/i]

Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

On a river with a current speed of \(3 \, \text{km/h}\), there are two harbors. Every Saturday, a cruise ship departs from Harbor 1 to Harbor 2, stays overnight, and returns to Harbor 1 on Sunday. On the ship live two snails, Romeo and Juliet. One Saturday, immediately after the ship departs, both snails start moving to meet each other and do so exactly when the ship arrives at Harbor 2. On the following Sunday, as the ship departs from Harbor 2, Romeo starts moving towards Juliet's house and reaches there exactly when the ship arrives back at Harbor 1. Given that Juliet moves half as fast as Romeo, determine the speed of the ship in still water. [i]Proposed by Demetre Gelashvili, Georgia [/i]

Peter and Basil take turns drawing roads on a plane, Peter starts. The road is either horizontal or a vertical line along which one can drive in only one direction (that direction is determined when the road is drawn). Can Basil always act in such a way that after each of his moves one could drive according to the rules between any two constructed crossroads, regardless of Peter's actions? Alexandr Perepechko

Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.

Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.

The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$. What is the least period of the function $\cos(\sin(x))$? $\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi\qquad\textbf{(D)}\ 4\pi\qquad\textbf{(E)}$ It's not periodic.