Find the smallest natural number that is the sum of $9$ consecutive natural numbers, is the sum of $10$ consecutive natural numbers and is also the sum of $11$ consecutive natural numbers.

Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths $6$, $8$, and $10$ and that no part of him is below the $x$-axis, find the minimum possible value of $k$.

Suppose \[ n(n\plus{}1)a_{n\plus{}1}\equal{}n(n\minus{}1)a_n\minus{}(n\minus{}2)a_{n\minus{}1} \] for every positive integer $ n\ge1$. Given that $ a_0\equal{}1,a_1\equal{}2$, find \[ \frac{a_0}{a_1}\plus{}\frac{a_1}{a_2}\plus{}\frac{a_2}{a_3}\plus{}\dots\plus{}\frac{a_{50}}{a_{51}}. \]

The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers $x$, $y$, and $z$ on the board, and does the following. First, $x$ is erased and replaced with $y$, after which $y$ is erased and replaced with $z$, and finally $z$ is erased and replaced with $x$. The higher power repeats this process some finite number of times. For example, if $(x,y,z)=(2,4,5)$ is chosen, followed by $(x,y,z)=(1,4,3)$, the board would change in the following manner: \[12345678 \rightarrow 14352678 \rightarrow 43152678\] Compute the number of possible final orderings of the eight numbers.

Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}$.

Consider parallelogram $ABCD$ with $AB > BC$. Point $E$ on $\overline{AB}$ and point $F$ on $\overline{CD}$ are marked such that there exists a circle $\omega_1$ passing through $A$, $D$, $E$, $F$ and a circle $\omega_2$ passing through $B$, $C$, $E$, $F$. If $\omega_1$, $\omega_2$ partition $\overline{BD}$ into segments $\overline{BX}$, $\overline{XY}$ , $\overline{Y D}$ in that order, with lengths $200$, $9$, $80$, respectively, compute $BC$.

There are $N$ acute triangles on the plane. Their vertices are all integer points, their areas are all equal to $2^{2020}$, but no two of them are congruent. Find the maximum possible value of $N$. Note: $(x,y)$ is an integer point if and only if $x$ and $y$ are both integers. [i]Proposed by CSJL[/i]

Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them.

Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.

An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes? ${{ \textbf{(A)}\ \frac{a}{1080r}\qquad\textbf{(B)}\ \frac{30r}{a}\qquad\textbf{(C)}\ \frac{30a}{r}\qquad\textbf{(D)}\ \frac{10r}{a} }\qquad\textbf{(E)}\ \frac{10a}{r} } $

Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+\frac1b$ and $b+\frac1a$ are different colors.

Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.

Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle. (A.Karlyuchenko)

Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$. [i]Proposed by Marcin E. Kuczma, Poland[/i]

Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$. [i]Proposed by Evan Chen[/i]

Some cells of a $2n\times2n$ board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the number of such markings.

For a real number $ x$ is $ [x]$ the next smaller integer to $ x$, that is the integer $ g$ with $ g\leqq<g+1$, and $ \{x\}=x-[x]$ is the “decimal part” of $ x$. Determine all triples $ (a,b,c)$ of real numbers, which fulfil the following system of equations: \[ \{a\}+[b]+\{c\}=2,9\]\[ \{b\}+[c]+\{a\}=5,3\]\[\{c\}+[a]+\{b\}=4,0\]

Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers. (a) Prove that there exist infinitely many sets of $4$ consecutive good numbers. (b) Find all sets of $5$ consecutive good numbers. [i]Proposed by Michael Ma[/i]

The square of an integer is called a [i]perfect square[/i]. If $x$ is a perfect square, the next larger perfect square is $\textbf{(A) }x+1\qquad\textbf{(B) }x^2+1\qquad\textbf{(C) }x^2+2x+1\qquad\textbf{(D) }x^2+x\qquad\textbf{(E) }x+2\sqrt{x}+1$

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

The price of 6 roses and 3 carnations is higher than 24 [i]yuan[/i], while the price of 4 roses and 5 carnations is lower than 22 [i]yuan[/i]. Compare the price of 2 roses and 3 carnations, the result is $\text{(A)}$ The price of 2 roses is higher. $\text{(B)}$ The price of 3 carnations is higher. $\text{(C)}$ Their prices are the same. $\text{(D)}$ Unknown.

In right triangle $ABC$, $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$. $M$ is the midpoint of $AB$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $CP$. Line $AB$ meets the circumcircle of $CNB$ again at $Q$. $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$, and $V$ lies on segment $SP$ such that $AV = VP$. Line $SP$ meets the circumcircle of $CPB$ again at $T$. $W$ lies on ray $VA$ past $A$ such that $2AW = ST$, and $O$ is the circumcenter of $SPM$. Prove that lines $OM$ and $BW$ are perpendicular.

Robin has obtained a circular pizza with radius $2$. However, being rebellious, instead of slicing the pizza radially, he decides to slice the pizza into $4$ strips of equal width both vertically and horizontally. What is the area of the smallest piece of pizza?

Let $ f$ be a function defined on $ \text{N}_0 \equal{} \{ 0,1,2,3,...\}$ and with values in $ \text{N}_0$, such that for $ n,m \in \text{N}_0$ and $ m \leq 9, f(10n \plus{} m) \equal{} f(n) \plus{} 11m$ and $ f(0) \equal{} 0.$ How many solutions are there to the equation $ f(x) \equal{} 1995$? A. None B. 1 C. 2 D. 11 E. Infinitely many

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.