A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$. [hide](It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )[/hide] M. Antipov

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.

Find all prime numbers $p$ and $q$ such that $2p^2q + 45pq^2$ is a perfect square.

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.

$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that $$ TB\cdot TC=TI_b^2.$$

Solve in positive numbers the system $ x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1$

Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.

Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

$ ABC$ and $ A'B'C'$ are two triangles in the same plane such that the lines $ AA',BB',CC'$ are mutually parallel. Let $ [ABC]$ denotes the area of triangle $ ABC$ with an appropriate $ \pm$ sign, etc.; prove that \[ 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].\]

Let $\mathcal{S}$ be a set of points in the plane such that for each subset $\mathcal{T}$ of $\mathcal{S}$, there exists a convex $2025$-gon which contains all of the points in $\mathcal{T}$ and none of the rest of the points in $\mathcal{S}$ but not $\mathcal{T}$. Determine the greatest possible number of points in $\mathcal{S}$. [i]Jason Lee[/i]

Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

Let $r_1, r_2, ..., r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute $$\sum^{47}_{i=1}r^{2020}_i .$$

A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?

For real numbers, $a,b,c$ with $bc \ne 0$ we have to $\frac{1-c^2}{bc} \ge 0$. Prove that $$5( a^2+b^2+c^2 -bc^3) \ge ab.$$

The following diagram shows four vertices connected by six edges. Suppose that each of the edges is randomly painted either red, white, or blue. The probability that the three edges adjacent to at least one vertex are colored with all three colors is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/6/4/de0a2a1a659011a30de1859052284c696822bb.png[/img]

Define a [i]growing spiral[/i] in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n\ge 2$ and: • The directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc. • The lengths of these line segments are positive and strictly increasing. \[\begin{picture}(200,180) \put(20,100){\line(1,0){160}} \put(100,10){\line(0,1){170}} \put(0,97){West} \put(180,97){East} \put(90,0){South} \put(90,180){North} \put(100,100){\circle{1}}\put(100,100){\circle{2}}\put(100,100){\circle{3}} \put(115,100){\circle{1}}\put(115,100){\circle{2}}\put(115,100){\circle{3}} \put(115,130){\circle{1}}\put(115,130){\circle{2}}\put(115,130){\circle{3}} \put(40,130){\circle{1}}\put(40,130){\circle{2}}\put(40,130){\circle{3}} \put(40,20){\circle{1}}\put(40,20){\circle{2}}\put(40,20){\circle{3}} \put(170,20){\circle{1}}\put(170,20){\circle{2}}\put(170,20){\circle{3}} \multiput(100,99.5)(0,.5){3}{\line(1,0){15}} \multiput(114.5,100)(.5,0){3}{\line(0,1){30}} \multiput(40,129.5)(0,.5){3}{\line(1,0){75}} \multiput(39.5,20)(.5,0){3}{\line(0,1){110}} \multiput(40,19.5)(0,.5){3}{\line(1,0){130}} \put(102,90){P0} \put(117,90){P1} \put(117,132){P2} \put(28,132){P3} \put(30,10){P4} \put(172,10){P5} \end{picture}\] How many of the points $(x,y)$ with integer coordinates $0\le x\le 2011,0\le y\le 2011$ [i]cannot[/i] be the last point, $P_n,$ of any growing spiral?

Let $P(x) = x^2 + ax + b$. The two zeros of $P$, $r_1$ and $r_2$, satisfy the equation $|r_1^2 -r_2^2| = 17$. Give that $a, b > 1$ and are both integers, find $P(1)$.

The plane angles at vertex $D$ of the pyramid $ABCD$ are equal to $\alpha$,$\beta$ and $\gamma$ ($\angle CDB = a$). An arbitrary point $M$ is taken on edge $CB$. A ball is inscribed in each of the pyramids $ABDM$ and $ACDM$. Let us draw through $D$ a plane distinct from $BCD$, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment $AM$ at point $P$. What is $\angle ADP$ equal to?

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.

The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.

A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as the product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?