A chess piece moves as follows: it can jump 8 or 9 squares either vertically or horizontally. It is not allowed to visit the same square twice. At most, how many squares can this piece visit on a $15 \times 15$ board (it can start from any square)? [i](4 points)[/i]

Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is $80$ cm tall, $120$ cm wide and its corners are denoted by $A, B, C$ and $D$ in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and $B$. The third vertex of the yellow foil is $S$, the trisecting point of side $DC$ closer to $D$, whereas the third vertex of the red foil is $P$, which is one fourth on the way on segment $SC$, closer to $C$. The red region (i.e. triangle $BPE$) is of area $16$ dm$^2$. What is the total area of the regions not covered by foil? [img]https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png[/img]

Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$

If $ x$, $ y$, and $ z$ are positive with $ xy \equal{} 24$, $ xz \equal{} 48$, and $ yz \equal{} 72$, then $ x \plus{} y \plus{} z$ is $ \textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24$

Spaceman Fred's spaceship (which has negligible mass) is in an elliptical orbit about Planet Bob. The minimum distance between the spaceship and the planet is $R$; the maximum distance between the spaceship and the planet is $2R$. At the point of maximum distance, Spaceman Fred is traveling at speed $v_\text{0}$. He then fires his thrusters so that he enters a circular orbit of radius $2R$. What is his new speed? [asy] size(300); // Shape draw(circle((0,0),25),dashed+gray); draw(circle((0,0),3.5),linewidth(2)); draw(ellipse((5,0),20,15)); // Dashed Lines draw((25,13)--(25,-35),dotted); draw((0,-35)--(0,-3.3),dotted); draw((0,3.3)--(0,13),dotted); draw((-15,13)--(-15,-35),dotted); // Labels draw((-14,-35)--(-1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"$R$",(-7.5,-35),N); draw((24,-35)--(1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"$2R$",(10,-35),N); // Blobs on Earth path A=(-1.433, 2.667)-- (-1.433, 2.573)-- (-1.360, 2.478)-- (-1.408, 2.360)-- (-1.493, 2.207)-- (-1.554, 2.160)-- (-1.614, 2.113)-- (-1.675, 2.065)-- (-1.735, 1.959)-- (-1.772, 1.877)-- (-1.723, 1.759)-- (-1.748, 1.676)-- (-1.748, 1.523)-- (-1.772, 1.369)-- (-1.760, 1.240)-- (-1.857, 1.145)-- (-1.941, 1.098)-- (-2.050, 1.122)-- (-2.111, 1.086)-- (-2.244, 1.039)-- (-2.390, 1.004)-- (-2.511, 0.909)-- (-2.486, 0.697)-- (-2.499, 0.555)-- (-2.535, 0.414)-- (-2.668, 0.308)-- (-2.765, 0.237)-- (-2.910, 0.131)-- (-3.068, 0.036)-- (-3.250, 0.024)-- (-3.310, 0.154)-- (-3.274, 0.272)-- (-3.286, 0.402)-- (-3.298, 0.532)-- (-3.250, 0.650)-- (-3.165, 0.768)-- (-3.128, 0.933)-- (-3.068, 1.074)-- (-3.032, 1.204)-- (-2.971, 1.310)-- (-2.886, 1.452)-- (-2.801, 1.558)-- (-2.729, 1.652)-- (-2.656, 1.770)-- (-2.583, 1.912)-- (-2.486, 1.995)-- (-2.365, 2.089)-- (-2.244, 2.207)-- (-2.123, 2.313)-- (-2.014, 2.419)-- (-1.905, 2.478)-- (-1.832, 2.573)-- (-1.687, 2.643)-- (-1.578, 2.714)--cycle; filldraw(A,gray); path B=(-0.397, 2.527)-- (-0.468, 2.321)-- (-0.538, 2.154)-- (-0.639, 2.065)-- (-0.760, 2.085)-- (-0.922, 2.085)-- (-0.993, 2.016)-- (-0.770, 1.918)-- (-0.649, 1.829)-- (-0.498, 1.780)-- (-0.367, 1.770)-- (-0.205, 1.751)-- (-0.084, 1.761)-- (-0.104, 1.613)-- (-0.114, 1.495)-- (-0.094, 1.358)-- (0.007, 1.220)-- (0.067, 1.131)-- (0.108, 1.013)-- (0.188, 0.905)-- (0.239, 0.787)-- (0.330, 0.650)-- (0.461, 0.620)-- (0.622, 0.620)-- (0.794, 0.591)-- (0.905, 0.610)-- (0.956, 0.689)-- (1.026, 0.591)-- (1.097, 0.483)-- (1.198, 0.374)-- (1.258, 0.276)-- (1.339, 0.188)-- (1.319, -0.009)-- (1.309, -0.166)-- (1.198, -0.343)-- (1.077, -0.432)-- (0.935, -0.520)-- (0.814, -0.589)-- (0.633, -0.677)-- (0.481, -0.727)-- (0.350, -0.776)-- (0.229, -0.894)-- (0.229, -1.041)-- (0.229, -1.228)-- (0.340, -1.346)-- (0.522, -1.415)-- (0.643, -1.513)-- (0.693, -1.651)-- (0.784, -1.798)-- (0.723, -1.936)-- (0.612, -2.044)-- (0.471, -2.123)-- (0.350, -2.201)-- (0.249, -2.270)-- (0.108, -2.339)-- (-0.013, -2.418)-- (-0.124, -2.535)-- (-0.135, -2.673)-- (-0.175, -2.811)-- (-0.084, -2.840)-- (0.067, -2.840)-- (0.209, -2.830)-- (0.350, -2.742)-- (0.522, -2.653)-- (0.582, -2.604)-- (0.713, -2.545)-- (0.845, -2.457)-- (0.935, -2.408)-- (1.057, -2.388)-- (1.228, -2.280)-- (1.329, -2.191)-- (1.460, -2.132)-- (1.581, -2.093)-- (1.692, -2.044)-- (1.793, -2.005)-- (1.844, -1.906)-- (1.844, -1.828)-- (1.904, -1.749)-- (2.005, -1.621)-- (1.955, -1.454)-- (1.894, -1.287)-- (1.773, -1.189)-- (1.632, -0.992)-- (1.592, -0.874)-- (1.491, -0.736)-- (1.410, -0.569)-- (1.460, -0.412)-- (1.561, -0.274)-- (1.592, -0.078)-- (1.622, 0.168)-- (1.551, 0.306)-- (1.440, 0.404)-- (1.420, 0.561)-- (1.551, 0.620)-- (1.703, 0.630)-- (1.824, 0.532)-- (1.955, 0.365)-- (2.046, 0.453)-- (2.116, 0.551)-- (2.167, 0.689)-- (2.096, 0.807)-- (1.965, 0.905)-- (1.834, 0.935)-- (1.743, 0.994)-- (1.622, 1.131)-- (1.531, 1.249)-- (1.430, 1.348)-- (1.359, 1.515)-- (1.420, 1.702)-- (1.511, 1.839)-- (1.571, 2.016)-- (1.672, 2.134)-- (1.592, 2.232)-- (1.440, 2.291)-- (1.289, 2.350)-- (1.178, 2.252)-- (1.127, 2.134)-- (1.067, 1.997)-- (0.986, 1.898)-- (0.845, 1.839)-- (0.693, 1.839)-- (0.522, 1.859)-- (0.471, 1.977)-- (0.380, 2.124)-- (0.289, 2.203)-- (0.188, 2.291)-- (0.047, 2.311)-- (-0.074, 2.370)-- (-0.195, 2.508)--cycle; filldraw(B,gray); [/asy] (A) $\sqrt{3/2}v_\text{0}$ (B) $\sqrt{5}v_\text{0}$ (C) $\sqrt{3/5}v_\text{0}$ (D) $\sqrt{2}v_\text{0}$ (E) $2v_\text{0}$

Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities \[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\] hold for all real numbers $X$. [i]Morteza Saghafian, Iran[/i]

A sequence of positive integer numbers $a_1,...,a_{100}$ such is that $a_1=1$, and for all $n=1, 2,...,100$ number $(a_1+...+a_n) \left ( \frac{1}{a_1}+...+\frac{1}{a_n} \right )$ is integer. What is the maximum value that can take $ a_{100}$? [i] M. Turevskii [/i]

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

I give you a deck of $n$ cards numbered $1$ through $n$. On each turn, you take the top card of the deck and place it anywhere you choose in the deck. You must arrange the cards in numerical order, with card $1$ on top and card $n$ on the bottom. If I place the deck in a random order before giving it to you, and you know the initial order of the cards, what is the expected value of the minimum number of turns you need to arrange the deck in order?

In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is: $\text{(A)}\ 16n\qquad \text{(B)}\ 18n\qquad \text{(C)}\ 21n\qquad \text{(D)}\ 24n\qquad \text{(E)}\ 27n$

Let $S{}$ be the set of nonnegative integers, which cointain only digits $0$ and $1$ in base $4$ numeral system. a) Show that if $x\in S, y\in S, x\neq y,$ then $\frac{x+y}{2}\notin S$. b) Let $T$ be a set of nonnegative integers such that $S\subset T, T\neq S$. Show that there exist $x\in T, y\in T, x\neq y,$ such that $\frac{x+y}{2} \in T$.

Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.) [list] [*] $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$ [*] $|A| = |B| \geq 1$ [*] Sum of elements in $A$ and sum of elements in $B$ are equal. [/list]

If the value of the infinite sum $$\frac{1}{2^2-1^2}+\frac{1}{4^2-2^2}+\frac{1}{8^2-4^2}+\frac{1}{16^2-8^2}+\dots.$$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]

find all real coefficient polynomial $ P(x)$ such that $ P(x)P(x\plus{}1)\equal{}P(x^2\plus{}x\plus{}1)$ for all $ x$

Find all relatively prime integers $a,b$ such that $$a^2+a=b^3+b$$

Consider the right isosceles $\vartriangle ABC$ at $ A$. Let $L$ be the intersection of the bisector of $\angle ACB$ with $AB$ and $K$ the intersection point of $CL$ with the bisector of $BC$. Let $X$ be the point on line $AK$ such that $\angle KCX = 90^o$ and let $Y$ be the point of intersection of $CX$ with the circumcircle of $\vartriangle ABC$. Let $Y'$ the reflection of point $Y$ wrt $BC$. Prove that $B - K -Y'$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

For $A\subset\mathbb Z$ and $a,b\in\mathbb Z$. We define $aA+b: =\{ax+b|x\in A\}$. If $a\neq0$ then we calll $aA+b$ and $A$ to similar sets. In this question the Cantor set $C$ is the number of non-negative integers that in their base-3 representation there is no $1$ digit. You see \[C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)\] (i.e. $C$ is partitioned to sets $3C$ and $3C+2$). We give another example $C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)$. A representation of $C$ is a partition of $C$ to some similiar sets. i.e. \[C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)\] and $C_{i}=a_{i}C+b_{i}$ are similar to $C$. We call a representation of $C$ a primitive representation iff union of some of $C_{i}$ is not a set similar and not equal to $C$. Consider a primitive representation of Cantor set. Prove that a) $a_{i}>1$. b) $a_{i}$ are powers of 3. c) $a_{i}>b_{i}$ d) (1) is the only primitive representation of $C$.

Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$. [i]Dan Schwartz[/i]

How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?

Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.

[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$. (Mikhail Plotnikov)

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the feet from $B$ and $C$ to the median from $A$, respectively. Suppose $DE=4$ and $CD=5$. Find the area of $ABC.$

Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$? $ \textbf{(A)}\ x^{1/6} \qquad \textbf{(B)}\ x^{1/4} \qquad \textbf{(C)}\ x^{3/8} \qquad \textbf{(D)}\ x^{1/2} \qquad \textbf{(E)}\ x$