In each field of a table there is a real number. We call such $n \times n$ table [i]silly[/i] if each entry equals the product of all the numbers in the neighbouring fields. a) Find all $2 \times 2$ silly tables. b) Find all $3 \times 3$ silly tables.

Find all real solutions of the equation $\sin^{5}x+\cos^{3}x=1$.

We call a positive integer $n$ $\textit{sixish}$ if $n=p(p+6)$, where $p$ and $p+6$ are prime numbers. For example, $187=11\cdot17$ is sixish, but $475=19\cdot25$ is not sixish. Define a function $f$ on the positive integers such that $f(n)$ is the sum of the squares of the positive divisors of $n$. For example, $f(10)=1^2+2^2+5^2+10^2=130$. (a) Find, with proof, an irreducible polynomial function $g(x)$ with integer coefficients such that $f(n)=g(n)$ for all sixish $n$. ("Irreducible" means that $g(x)$ cannot be factored as the product of two polynomials of smaller degree with integer coefficients.) (b) We call a positive integer $n$ $\textit{pseudo-sixish}$ if $n$ is not sixish but nonetheless $f(n)=g(n)$, where $g(n)$ is the polynomial function that you found in part (a). Find, with proof, all pseudo-sixish positive integers.

Enzymes convert glucose $(M=180.2)$ to ethanol $(M=46.1)$ according to the equation \[ \text{C}_6\text{H}_{12}\text{O}_6 \rightarrow 2\text{C}_2\text{H}_2\text{OH} + 2\text{CO}_2 \] What is the maximum mass of ethanol that can be made from $15.5$ kg of glucose? $ \textbf{(A)}\hspace{.05in}0.256 \text{kg}\qquad\textbf{(B)}\hspace{.05in}0.512 \text{kg} \qquad\textbf{(C)}\hspace{.05in}3.96 \text{kg}\qquad\textbf{(D)}\hspace{.05in}7.93 \text{kg}\qquad$

$\dfrac{1}{167}$ has a period of length $166$, and its first $83$ digits after the decimal point are: \begin{align*} &.0059880239 \hspace{0.1 cm} 5209580838 \hspace{0.1 cm} 3233532934 \hspace{0.1 cm} 1317365269 \\ &\hspace{0.1 cm} 4610778443 \hspace{0.1 cm} 1137724550 \hspace{0.1 cm} 8982035928 \hspace{0.1 cm} 1437125748\\ &\hspace{0.1 cm} 502 \end{align*} Let $a$, $b$, $c$ be the $120$th, $121$st, $122$nd digits after the decimal point of $\dfrac{1}{167}$, respectively. Find $100a + 10b + c$. [i]Lightning 4.2[/i]

The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.

Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D`, F`, G`$ and $E`$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF`$ with $EG`$, $Y$ is the intersection of $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ and $W$ is the intersection of $EF`$ with $DG`$. Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.

Investigate the singular points of the differential form $dt = dx/y$ on the compact Riemann surface $y^2/2 + U(x) = E$, where $U$ is a polynomial and $E$ is not a critical value.

In the equilateral triangle $ABC$ ($AB = 2$), cevians are drawn that do not intersect at one point. It turned out that the pairwise intersection points of these cevians lie on the inscribed circle of triangle $ABC$. Find the length of the cevian segment from the vertex of the triangle to the nearest point of intersection with the circle.

[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url] [url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square. [url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$. [url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value. [url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

In a right-angled triangle $ABC$ with the hypotenuse $BC$, $D$ is the foot of the altitude from $A$. The line through the incenters of the triangles $ABD$ and $ADC$ intersects the legs of $\triangle ABC$ at $E$ and $F$. Prove that $A$ is the circumcenter of triangle $DEF$.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

The sum of the first $n$ terms of the sequence \[1,~(1+2),~(1+2+2^2),~\dots ~(1+2+2^2+\dots +2^{n-1})\] in terms of $n$ is $\textbf{(A) }2^n\qquad\textbf{(B) }2^n-n\qquad\textbf{(C) }2^{n+1}-n\qquad\textbf{(D) }2^{n+1}-n-2\qquad \textbf{(E) }n\cdot 2^n$

Let there be given three circles $K_1,K_2,K_3$ with centers $O_1,O_2,O_3$ respectively, which meet at a common point $P$. Also, let $K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}$. Given an arbitrary point $X$ on $K_1$, join $X$ to $A$ to meet $K_2$ again in $Y$ , and join $X$ to $C$ to meet $K_3$ again in $Z.$ [b](a)[/b] Show that the points $Z,B, Y$ are collinear. [b](b)[/b] Show that the area of triangle $XY Z$ is less than or equal to $4$ times the area of triangle $O_1O_2O_3.$

Consider a positive integer $n=\overline{a_1a_2...a_k},k\ge 2$.A [i]trunk[/i] of $n$ is a number of the form $\overline{a_1a_2...a_t},1\le t\le k-1$.(For example,the number $23$ is a [i]trunk[/i] of $2351$.) By $T(n)$ we denote the sum of all [i]trunk[/i] of $n$ and let $S(n)=a_1+a_2+...+a_k$.Prove that $n=S(n)+9\cdot T(n)$.

We say $m \circ n$ for natural m,n $\Longleftrightarrow$ nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$ We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$ $\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$ Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.

Karlson and Smidge divide a cake in a shape of a square in the following way. First, Karlson places a candle on the cake (chooses some interior point). Then Smidge makes a straight cut from the candle to the boundary in the direction of his choice. Then Karlson makes a straight cut from the candle to the boundary in the direction perpendicular to Smidge's cut. As a result, the cake is split into two pieces; Smidge gets the smaller one. Smidge wants to get a piece which is no less than a quarter of the cake. Can Karlson prevent Smidge from getting the piece of that size?

Let $O$ be the circumcenter of triangle $ABC$, and $\omega$ be the circumcircle of triangle $BOC$. Line $AO$ intersects with circle $\omega$ again at the point $G$. Let $M$ be the midpoint of side $BC$, and the perpendicular bisector of $BC$ meets circle $\omega$ at the points $O$ and $N$. Prove that the midpoint of the segment $AN$ lies on the radical axis of the circumcircle of triangle $OMG$, and the circle whose diameter is $AO$.

[b]p1.[/b] Find the smallest positive integer $N$ such that $2N -1$ and $2N +1$ are both composite. [b]p2.[/b] Compute the number of ordered pairs of integers $(a, b)$ with $1 \le a, b \le 5$ such that $ab - a - b$ is prime. [b]p3.[/b] Given a semicircle $\Omega$ with diameter $AB$, point $C$ is chosen on $\Omega$ such that $\angle CAB = 60^o$. Point $D$ lies on ray $BA$ such that $DC$ is tangent to $\Omega$. Find $\left(\frac{BD}{BC} \right)^2$. [b]p4.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If $f(x)$ is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p5.[/b] Regular hexagon $ABCDEF$ has side length $3$. Circle $\omega$ is drawn with $AC$ as its diameter. $BC$ is extended to intersect $\omega$ at point $G$. If the area of triangle $BEG$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree and $gcd(a, c) = 1$, find $a + b + c$. [b]p6.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p7.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p8.[/b] Let $\vartriangle ABC$ be an equilateral triangle. Isosceles triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, not overlapping $\vartriangle ABC$, are constructed such that each has area seven times the area of $\vartriangle ABC$. Compute the ratio of the area of $\vartriangle DEF$ to the area of $\vartriangle ABC$. [b]p9.[/b] Consider the sequence of polynomials an(x) with $a_0(x) = 0$, $a_1(x) = 1$, and $a_n(x) = a_{n-1}(x) + xa_{n-2}(x)$ for all $n \ge 2$. Suppose that $p_k = a_k(-1) \cdot a_k(1)$ for all nonnegative integers $k$. Find the number of positive integers $k$ between $10$ and $50$, inclusive, such that $p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}$. [b]p10.[/b] In triangle $ABC$, point $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle BEC has area 45 and triangle $ADC$ has area $72$, and lines CH and AB meet at F. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with c squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p11.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer x such that $x^2 + 18x + y$ is a perfect fourth power. [b]p12.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a$, $b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Let $a,b,c \ge 0$ and $a+b+c=3$. Prove that: $2(ab+bc+ca)-3abc\ge \sum_{cyc}^{}a\sqrt{\frac{b^2+c^2}{2}}$

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.

Let $p>10$ be a prime. Prove that there is positive integers $m,n$ with $m+n<p$ such that $p$ divides $5^m7^n -1$