Define a search algorithm called $\texttt{powSearch}$. Throughout, assume $A$ is a 1-indexed sorted array of distinct integers. To search for an integer $b$ in this array, we search the indices $2^0,2^1,\ldots$ until we either reach the end of the array or $A[2^k] > b$. If at any point we get $A[2^k] = b$ we stop and return $2^k$. Once we have $A[2^k] > b > A[2^{k-1}]$, we throw away the first $2^{k-1}$ elements of $A$, and recursively search in the same fashion. For example, for an integer which is at position $3$ we will search the locations $1, 2, 4, 3$. Define $g(x)$ to be a function which returns how many (not necessarily distinct) indices we look at when calling $\texttt{powSearch}$ with an integer $b$ at position $x$ in $A$. For example, $g(3) = 4$. If $A$ has length $64$, find \[g(1) + g(2) + \ldots + g(64).\]

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

For a positive integer $n$, $n$ vertices which have $10000$ written on them exist on a plane. For $3$ vertices that are collinear and are written positive numbers on them, denote procedure $P$ as subtracting $1$ from the outer vertices and adding $2023$ to the inner vertical. Show that procedure $P$ cannot be repeated infinitely.

You are given a $5\times 6$ checkerboard with squares alternately shaded black and white. The bottom- left square is white. Each square has side length $1$ unit. You can normally travel on this board at a speed of $2$ units per second, but while you travel through the interior (not the boundary) of a black square, you are slowed down to $1$ unit per second. What is the shortest time it takes to travel from the bottom-left corner to the top-right corner of the board?

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.

Let $H$ be the orthocenter of the triangle $ABC$ and let $D$, $E$, $F$ be the feet of heights by $A$, $B$, $C$. Let $\omega_D$, $\omega_E$, $\omega_F$ be the incircles of $FEH$, $DHF$, $HED$ and let $I_D$, $I_E$, $I_F$ be their centers. Show that $I_DD$, $I_EE$ and $I_FF$ compete.

An integer $ m > 1$ is given. The infinite sequence $ (x_n)_{n\ge 0}$ is defined by $ x_i\equal{}2^i$ for $ i<m$ and $ x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m}$ for $ i\ge m$. Find the greatest natural number $ k$ such that there exist $ k$ successive terms of this sequence which are divisible by $ m$.

Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

A man who is $2$ meters tall is standing $5$ meters away from a lamppost that is $6$ meters high. How long is the man's shadow cast by the lamppost, in meters? $\text{(A) }2\qquad\text{(B) }\frac{7}{3}\qquad\text{(C) }\frac{5}{2}\qquad\text{(D) }4\qquad\text{(E) }\frac{5}{3}$

[b]p1.[/b] The serpent of fire and the serpent of ice play a game. Since the serpent of ice loves the lucky number $6$, he will roll a fair $6$-sided die with faces numbered $1$ through $6$. The serpent of fire will pay him $\log_{10} x$, where $x$ is the number he rolls. The serpent of ice rolls the die $6$ times. His expected total amount of winnings across the $6$ rounds is $k$. Find $10^k$. [b]p2.[/b] Let $a = \log_3 5$, $b = \log_3 4$, $c = - \log_3 20$, evaluate $\frac{a^2+b^2}{a^2+b^2+ab} +\frac{b^2+c^2}{b^2+c^2+bc} +\frac{c^2+a^2}{c^2+a^2+ca}$. [b]p3.[/b] Let $\vartriangle ABC$ be an isosceles obtuse triangle with $AB = AC$ and circumcenter $O$. The circle with diameter $AO$ meets $BC$ at points $X, Y$ , where X is closer to $B$. Suppose $XB = Y C = 4$, $XY = 6$, and the area of $\vartriangle ABC$ is $m\sqrt{n}$ for positive integers $m$ and $n$, where $n$ does not contain any square factors. Find $m + n$. [b]p4.[/b] Alice is not sure what to have for dinner, so she uses a fair $6$-sided die to decide. She keeps rolling, and if she gets all the even numbers (i.e. getting all of $2, 4, 6$) before getting any odd number, she will reward herself with McDonald’s. Find the probability that Alice could have McDonald’s for dinner. [b]p5.[/b] How many distinct ways are there to split $50$ apples, $50$ oranges, $50$ bananas into two boxes, such that the products of the number of apples, oranges, and bananas in each box are nonzero and equal? [b]p6.[/b] Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices $(1, 1)$,$(n, n)$ for some constant $n$. Sujay loses when the two-point pattern $P$ below shows up:[img]https://cdn.artofproblemsolving.com/attachments/1/9/d1fe285294d4146afc0c7a2180b15586b04643.png[/img] That is, Sujay loses when there exists a pair of points $(x, y)$ and $(x + 2, y + 1)$. He and Rishabh stop marking points when the pattern $P$ appears on the board. If Rishabh goes first, let $S$ be the set of all integers $3 \le n \le 100$ such that Rishabh has a strategy to always trick Sujay into being the one who creates $P$. Find the sum of all elements of $S$. [b]p7.[/b] Let $a$ be the shortest distance between the origin $(0, 0)$ and the graph of $y^3 = x(6y -x^2)-8$. Find $\lfloor a^2 \rfloor $. ($\lfloor x\rfloor $ is the largest integer not exceeding $x$) [b]p8.[/b] Find all real solutions to the following equation: $$2\sqrt2x^2 + x -\sqrt{1 - x^2 } -\sqrt2 = 0.$$ [b]p9.[/b] Given the expression $S = (x^4 - x)(x^2 - x^3)$ for $x = \cos \frac{2\pi}{5 }+ i\sin \frac{2\pi}{5 }$, find the value of $S^2$ . [b]p10.[/b] In a $32$ team single-elimination rock-paper-scissors tournament, the teams are numbered from $1$ to $32$. Each team is guaranteed (through incredible rock-paper-scissors skill) to win any match against a team with a higher number than it, and therefore will lose to any team with a lower number. Each round, teams who have not lost yet are randomly paired with other teams, and the losers of each match are eliminated. After the $5$ rounds of the tournament, the team that won all $5$ rounds is ranked $1$st, the team that lost the 5th round is ranked $2$nd, and the two teams that lost the $4$th round play each other for $3$rd and $4$th place. What is the probability that the teams numbered $1, 2, 3$, and $4$ are ranked $1$st, 2nd, 3rd, and 4th respectively? If the probability is $\frac{m}{n}$ for relatively prime integers $m$ and $n$, find $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares. b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.

Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$

What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

Numbers from $1$ to $1000$ are arranged around a circle. Prove that it is possible to form $500$ non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than $749$. (AA Razborov, Moscow)

The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$. [img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]

Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?

Let $P_1$, $P_2$, $\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.

In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.

Regular hexagon $ABCDEF$ with side length $2$ is inscribed within a sphere of radius $4$. Let point $X$ be on the sphere. Find the maximum value of the volume of the pyramid $ABCDEFX$.

Let $n$ be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular $2n$-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let $P(n)$ denote the number of different ways Luca can make $2n$ jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.) [b]a)[/b] Find the value of $P(n)$ if $n$ is odd. [b]b)[/b] Prove that if $n$ is even, then \[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\] [i]Proposed by Péter Csikvári and Kartal Nagy, Budapest[/i]

Find all solutions $(x,y,z)$ to the system $$\begin{cases}(x^2 - 6x + 13)y = 20 \\ (y^2 - 6y + 13)z = 20 \\ (z^2 - 6z + 13)x = 20 \end{cases}$$

Let $ a\in \mathbb{R} $ and $ f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R} $ are the additive functions such that for every $ x\in \mathbb{R} $ we have $ f_1(x)f_2(x) \cdots f_n(x) =ax^n $. Show that there exists $ b\in \mathbb {R} $ and $ i\in {\{1,2,\ldots,n}\} $ such that for every $ x\in \mathbb{R} $ we have $ f_i(x)=bx $.

Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.