Round to the nearest tenth: $\log_6 (6^2-6+1) + 3\log_6 (5) - \frac{1}{2}\log_6 (9)$.

What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?

Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\cdot AM$ is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair A = (-1,0); pair B = (1,0); pair C = (0,1); pair D = (0,-1); pair M = dir(45); pair P = intersectionpoint(O--C,A--M); draw(Circle(O,1)); draw(A--B); draw(C--D); draw(A--M); label("$A$",A,W); label("$B$",B,E); label("$C$",C,N); label("$D$",D,S); label("$M$",M,NE); label("$O$",O,NE); label("$P$",P,NW);[/asy]$ \textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \textbf{(C)}\ CP\cdot CD \qquad \textbf{(D)}\ CP\cdot PD\qquad$ $ \textbf{(E)}\ CO\cdot OP$

Prove that if $p$ and $p+2$ are prime integers greater than 3, then 6 is a factor of $p+1$.

[b]p1.[/b] $p, q, r$ are prime numbers such that $p^q + 1 = r$. Find $p + q + r$. [b]p2.[/b] $2014$ apples are distributed among a number of children such that each child gets a different number of apples. Every child gets at least one apple. What is the maximum possible number of children who receive apples? [b]p3.[/b] Cathy has a jar containing jelly beans. At the beginning of each minute he takes jelly beans out of the jar. At the $n$-th minute, if $n$ is odd, he takes out $5$ jellies. If n is even he takes out $n$ jellies. After the $46$th minute there are only $4$ jellies in the jar. How many jellies were in the jar in the beginning? [b]p4.[/b] David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work? [b]p5.[/b] Let $a, b, c, d$ be positive real numbers such that $$a^2 + b^2 = 1$$ $$c^2 + d^2 = 1;$$ $$ad - bc =\frac17$$ Find $ac + bd$. [b]p6.[/b] Three circles $C_A,C_B,C_C$ of radius $1$ are centered at points $A,B,C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap. [b]p7.[/b] Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\{3, 8, 16, 18, 24\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$? [b]p8.[/b] Let $S$ be the set of integers $1 \le N \le 2^{20}$ such that $N = 2^i + 2^j$ where $i, j$ are distinct integers. What is the probability that a randomly chosen element of $S$ will be divisible by $9$? [b]p9.[/b] Given a two-pan balance, what is the minimum number of weights you must have to weigh any object that weighs an integer number of kilograms not exceeding $100$ kilograms? [b]p10.[/b] Alex, Michael and Will write $2$-digit perfect squares $A,M,W$ on the board. They notice that the $6$-digit number $10000A + 100M +W$ is also a perfect square. Given that $A < W$, find the square root of the $6$-digit number. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alice and Bob are going to play a game called extra tricky double rock paper scissors (ETDRPS). In ETDRPS, each player simultaneously selects [i]two [/i] moves, one for his or her right hand, and one for his or her left hand. Whereas Alice can play rock, paper, or scissors, Bob is only allowed to play rock or scissors. After revealing their moves, the players compare right hands and left hands separately. Alice wins if she wins [i]strictly [/i] more hands than Bob. Otherwise, Bob wins. For example, if Alice and Bob were to both play rock with their right hands and scissors with their left hands, then both hands would be tied, so Bob would win the game. However, if Alice were to instead play rock with both hands, then Alice would win the left hand. The right hand would still be tied, so Alice would win the game. Assuming both players play optimally, compute the probability that Alice will win the game.

Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.

Solve the equation $m^{gcd(m,n)} = n^{lcm(m,n)}$ in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.

A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.

a) Is it possible to arrange numbers $1,2,...,13$ in a circumference such that the sum of any two neighbouring numbers to be a prime number? b) Is the same problem possible for the numbers $1,2,...,16$?

Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence. [i]Proposed by Sammy Charney[/i]

If a dealer could get his goods for $8\%$ less while keeping his selling price fixed, his profit, based on cost, would be increased to $(x+10)\%$ from his present profit of $x\%$, which is $\textbf{(A) }12\%\qquad\textbf{(B) }15\%\qquad\textbf{(C) }30\%\qquad\textbf{(D) }50\%\qquad \textbf{(E) }75\%$

The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.

In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle.

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

Consider an acute-angled triangle $ABC$ with $AB=AC$ and $\angle A>60^\circ$. Let $O$ be the circumcenter of $ABC$. Point $P$ lies on circumcircle of $BOC$ such that $BP\parallel AC$ and point $K$ lies on segment $AP$ such that $BK=BC$. Prove that $CK$ bisects the arc $BC$ of circumcircle of $BOC$.

Point $D$ lies on side $AB$ of triangle $ABC$. Let $\omega_1$ and $\Omega_1,\omega_2$ and $\Omega_2$ be the incircles and the excircles (touching segment $AB$) of triangles $ACD$ and $BCD.$ Prove that the common external tangents to $\omega_1$ and $\omega_2,$ $\Omega_1$ and $\Omega_2$ meet on $AB$.

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Circles $\omega_1$ and $\omega_2$ intersect each other at points $A$ and $B$. Point $C$ lies on the tangent line from $A$ to $\omega_1$ such that $\angle ABC = 90^\circ$. Arbitrary line $\ell$ passes through $C$ and cuts $\omega_2$ at points $P$ and $Q$. Lines $AP$ and $AQ$ cut $\omega_1$ for the second time at points $X$ and $Z$ respectively. Let $Y$ be the foot of altitude from $A$ to $\ell$. Prove that points $X, Y$ and $Z$ are collinear. [i]Proposed by Iman Maghsoudi[/i]

There is a board with three rows and $2023$ columns. In the first row the numbers are written from $1$ to $2023$, ordered from least to greatest. The devil writes those same numbers in the boxes in the second row, but ordered to his choice. Then, in each box in the third row he writes the difference between the two numbers already written in his own column (the largest minus the smallest). For example, if the first two boxes of a column are the numbers $21$ and $198$, in the third box it is written $198-21 = 177$. Explain why, no matter how the devil completed the second row of the board, it will never happen that multiplying them $2023$ numbers in the third row the result is odd.

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.

Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is [asy] size(180); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3; D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));[/asy] $\textbf{(A)}\ \dfrac{1}{6}\qquad \textbf{(B)}\ \dfrac{1}{4} \qquad \textbf{(C)}\ \dfrac{1}{3} \qquad \textbf{(D)}\ \dfrac{2}{5} \qquad \textbf{(E)}\ \dfrac{1}{2}$

Each cell of a $10 \times 10$ board is painted red, blue or white, with exactly twenty of them red. No two adjacent cells are painted in the same colour. A domino consists of two adjacent cells, and it is said to be good if one cell is blue and the other is white. (a) Prove that it is always possible to cut out $30$ good dominoes from such a board. (b) Give an example of such a board from which it is possible to cut out $40$ good dominoes. (c) Give an example of such a board from which it is not possible to cut out more than $30$ good dominoes.

In $\triangle ABC$, point $M$ is the middle point of $AC$. $MD//AB$ and meet the tangent of $A$ to $\odot(ABC)$ at point $D$. Point $E$ is in $AD$ and point $A$ is the middle point of $DE$. $\{P\}=\odot(ABE)\cap AC,\{Q\}=\odot(ADP)\cap DM$. Prove that $\angle QCB=\angle BAC$. [url=https://imgtu.com/i/4pZ7Zj][img]https://z3.ax1x.com/2021/09/12/4pZ7Zj.jpg[/img][/url]