A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice? [img]https://cdn.artofproblemsolving.com/attachments/e/c/c817d92ead6cfb3868f9cb526fb4e1fd7ffe4d.png[/img]

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w,v+4w)$ with $0 \le u \le 1$, $0 \le v \le 1$, and $0 \le w \le 1$? \\ \\ $\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.

There are $2020$ positive integers written on a blackboard. Every minute, Zuming erases two of the numbers and replaces them by their sum, difference, product, or quotient. For example, if Zuming erases the numbers $6$ and $3$, he may replace them with one of the numbers in the set $\{6+3, 6-3, 3-6, 6\times 3, 6\div 3, 3\div 6\}$ $= \{9, 3, 3, 18, 2, \tfrac 12\}$. After $2019$ minutes, Zuming writes the single number $-2020$ on the blackboard. Show that it was possible for Zuming to have ended up with the single number $2020$ instead, using the same rules and starting with the same $2020$ integers. [i]Proposed by Zhuo Qun (Alex) Song[/i]

25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is its speed $v_2$ when it passes the perigee? A) $v_2 = \frac{3}{\sqrt{5}}v_1$ B) $v_2 = \frac{3}{\sqrt{7}}v_1$ C) $v_2 = \frac{2}{\sqrt{3}}v_1$ D) $v_2 = \frac{\sqrt{7}}{\sqrt{3}}v_1$ E) $v_2 = 4v_1$

Find the sum of the squares of the roots of $x^2-5x-7$.

We say that a positive integer $n$ is $m$[i]-expressible[/i] if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$. Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible? [i]Proposed by Krit Boonsiriseth[/i]

A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.

The house SEAT recommends to the users, for the correct conservation of the wheels, periodic replacements of the same in the form $R \to 3 \to 2 \to 1 \to 4 \to R$, according to the numbering of the figure. Calling $G$ to this change of wheels, $G^2 = GG$ to making this change twice, and so on for the other powers of $G$, a) Show that the set of these powers forms a group, and study it. b) Each puncture of one of the wheels is also equivalent to a substitution in which said wheel is replaced by the spare one $ (R)$ and, once repaired, it comes to occupy the place of this obtained $G$ as a product of prick transformations. Do they form a group? [img]https://cdn.artofproblemsolving.com/attachments/4/a/712fede88321c67753417fda828a08ba528b4f.png[/img]

Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$.

Given a square-free integer $n>2$, evaluate the sum $\sum_{k=1}^{(n-2)(n-1)} \lfloor ({kn})^{1/3} \rfloor$.

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

We call a set $A\subset \mathbb{R}$ [i]free of arithmetic progressions[/i] if for all distinct $a,b,c\in A$ we have $a+b\neq 2c.$ Prove that the set $\{0,1,2,\ldots 3^8-1\}$ has a subset $A$ which is free of arithmetic progressions and has at least $256$ elements.

Let $a, b, c, d$ be the lengths of consecutive sides of a quadrilateral, and $S$ its area. Prove that $S \le \frac{ (a + b)(c + d)}{4}$

Let $ABC$ be a scalene triangle with incenter $I$ and symmedian point $K$. Furthermore, suppose that $BC = 1099$. Let $P$ be a point in the plane of triangle $ABC$, and let $D$, $E$, $F$ be the feet of the perpendiculars from $P$ to lines $BC$, $CA$, $AB$, respectively. Let $M$ and $N$ be the midpoints of segments $EF$ and $BC$, respectively. Suppose that the triples $(M,A,N)$ and $(K,I,D)$ are collinear, respectively, and that the area of triangle $DEF$ is $2020$ times the area of triangle $ABC$. Compute the largest possible value of $\lceil AB+AC\rceil$. [i]Proposed by Brandon Wang[/i]

Let $ABCD$ be a cyclic quadrilateral.$E$ is the midpoint of $(AC)$ and $F$ is the midpoint of $(BD)$ {$G$}=$AB\cap CD$ and {$H$}=$AD\cap BC$. a)Prove that the intersections of the angle bisector of $\angle{AHB}$ and the sides $AB$ and $CD$ and the intersections of the angle bisector of$\angle{AGD}$ with $BC$ and $AD$ are the verticles of a rhombus b)Prove that the center of this rhombus lies on $EF$

In a triangle $ABC$ with $AB<AC$, $D$ is a point on segment $AC$ such that $BD = CD$. A line parallel to $BD$ meets segment $BC$ at $E$ and line $AB$ at $F$. Point $G$ is the intersection of $AE$ and $BD$. Show that $\angle BCG = \angle BCF$. [i](Côte d’Ivoire)[/i]

(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$. (b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.

Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).

The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality \[ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. \]