Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $ [i]Gheorghe Iurea[/i]

I start with a sequence of letters $A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021}$. I go through $i = 1, 2, 3, \cdots, 6062$ in order, and for each $i$, I can choose to swap letters $i$ and $i+1$. Let $N$ be the number of distinct strings I can end up with. What is the remainder when $N$ is divided by $2017$?

What is the largest positive integer which is equal to the sum of its digits? [i]Proposed by Evan Chen[/i]

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 $ABCDEFX$.