Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.

An evil witch is making a potion to poison the people of PUMAClandia. In order for the potion to work, the number of poison dart frogs cannot exceed $5$, the number of wolves’ teeth must be an even number, and the number of dragon scales has to be a multiple of $6$. She can also put in any number of tiger nails. Given that the stew has exactly $2021$ ingredients, in how many ways can she add ingredients for her potion to work?

Given are the sequences \[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\] of positive real numbers. For each integer $n$ the following inequalities hold: \[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\] \[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\] \[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\] Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

[u]Round 5[/u] [b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle. [b]5.2.[/b] A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus? [b]5.3.[/b] A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is $\frac23$ , the volume of the combined cup and lid is $296\pi$, and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container? [u]Round 6[/u] [i]Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.[/i] [b]6.1.[/b] Let the answer to problem $2$ be $b$. There are b people in a room, each of which is either a truth-teller or a liar. Person $1$ claims “Person $2$ is a liar,” Person $2$ claims “Person $3$ is a liar,” and so on until Person $b$ claims “Person $1$ is a liar.” How many people are truth-tellers? [b]6.2.[/b] Let the answer to problem $3$ be $c$. What is twice the area of a triangle with coordinates $(0, 0)$, $(c, 3)$ and $(7, c)$ ? [b]6.3.[/b] Let the answer to problem $ 1$ be $a$. Compute the smaller zero to the polynomial $x^2 - ax + 189$ which has $2$ integer roots. [u]Round 7[/u] [b]7.1. [/b]Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states: [i]Every minute, either $\left\lfloor \frac{1000}{d} \right\rfloor$ or $\left\lceil \frac{1000}{d} \right\rceil$ kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.[/i] Over the next minute, $n$ kiwis fall on Neeton’s head. Let $S$ be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that $m < x < M$ for all elements $x$ in $S$. If the least possible value of $M - m$ is $\frac{2000}{16899}$ centimeters, what is the value of $n$? Note that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the least integer greater than or equal to $x$. [b]7.2.[/b] Nithin is playing chess. If one queen is randomly placed on an $ 8 \times 8$ chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.) [b]7.3.[/b] Nithin is writing binary strings, where each character is either a $0$ or a $1$. How many binary strings of length $12$ can he write down such that $0000$ and $1111$ do not appear? [u]Round 8[/u] [b]8.[/b] What is the period of the fraction $1/2018$? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2765571p24215461]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Through the points with integer coordinates in the right-angled coordinate system $Oxyz$ are constructed planes, parallel to the coordinate planes and in this way the space is divided to unit cubes. Find all triples ($a, b, c$) consisting of natural numbers ($a \le b \le c$) for which the cubes formed can be coloured in $abc$ colors in such a way that every palellepiped with dimensions $a \times  b \times c$, having vertices with integer coordinates and sides parallel to the coordinate axis doesn't contain unit cubes in the same color.

Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point? [i]A. Chukhnov[/i]

If two altitudes of a triangle have length $12$ and $4$, what integral lengths can the third altitude attain?

Determine the highest possible value of: $$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$ where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$. Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.

Let be three real numbers $ a,b,c\in [0,1] $ satisfying the condition $ ab+bc+ca=1. $ Prove that $$ a^2+b^2+c^2\le 2, $$ and determine the cases in which equality is attained.

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.

Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.

Solve in natural numbers: $$x^3-y^3=xy+61$$

Given a quadrilateral $ABCD$ with $\angle ABC = \angle ADC$. Let $BM$ be the altitude of the triangle $ABC$, and $M$ belongs to $AC$. Point $M'$ is marked on the diagonal $AC$ so that $$\frac{AM \cdot CM'}{ AM' \cdot CM}= \frac{AB \cdot CD }{ BC \cdot AD}$$ Prove that the intersection point of $DM'$ and $BM$ coincides with the orthocenter of the triangle $ABC$. (M. Zhikhovich)

\[\int_0^\infty e^{-\lfloor x \rfloor(1+\{x\})}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.

Let $\vartriangle ABC$ be a triangle, and let $C_0, B_0$ be the feet of perpendiculars from $C$ and $B$ onto $AB$ and $AC$ respectively. Let $\Gamma$ be the circumcircle of $\vartriangle ABC$. Let E be a point on the $\Gamma$ such that $AE \perp BC$. Let $M$ be the midpoint of $BC$ and let $G$ be the second intersection of EM and $\Gamma$. Let $T$ be a point on $\Gamma$ such that $T G$ is parallel to $BC$. Prove that $T, A, B_0, C_0$ are concyclic.

A triangle \( ABC \) is given with centers \( O \) and \( I \) of the circumscribed and inscribed circles, respectively. Point \( A_1 \) is the reflection of \( A \) with respect to \( I \). Point \( A_2 \) is such that lines \( BA_1 \) and \( BA_2 \) are symmetric with respect to \( BI \), and lines \( CA_1 \) and \( CA_2 \) are symmetric with respect to \( CI \). Prove that \( AO^2 = |A_2O^2 - A_2I^2| \).

Suppose that $a$, $b$ and $x$ are positive real numbers. Prove that $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$.

Trapezoid $ABCD,$ with $AB \parallel CD,$ has side lengths $AB=11, BC=8, CD=19,$ and $DA=4.$ Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle{ABC}, \triangle{BCD}, \triangle{CDA},$ and $\triangle{DAB}.$

Evaluate $1 + 2 + 4 + 7$ $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

A can contains a mixture of two liquids A and B in the ratio $7:5$. When $9$ litres of the mixture are drawn and replaced by the same amount of liquid $B$, the ratio of $A$ and $B$ becomes $7:9$. How many litres of liquid A was contained in the can initially? $\textbf{(A)}~18$ $\textbf{(B)}~19$ $\textbf{(C)}~20$ $\textbf{(D)}~\text{None of the above}$