Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$? $ \textbf{(A) }0 \qquad \textbf{(B) }2 \qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

Kate has two bags $X$ and $Y$. Bag $X$ contains $5$ red marbles (and nothing else). Bag $Y$ contains $4$ red marbles and $1$ blue marble (and nothing else). Kate chooses one of her bags at random (each with probability $\frac{1}{2}$) and removes a random marble from that bag (each marble in that bag being equally likely). She repeats the previous step until one of the bags becomes empty. At that point, what is the probability that the blue marble is still in bag $Y$?

Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

The number $2.5081081081081\ldots$ can be written as $\frac{m}{n}$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.

[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions. [b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale. [b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number? [b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it? [b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. If $n \geq 5$ and $e(G_{n}) \geq \frac{n^{2}}{4}+2$, prove that $G_{n}$ contains two triangles that share exactly one vertex.

How many four digit perfect square numbers are there in the form $AABB$ where $A,B \in \{1,2,\dots, 9\}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None of above} $

Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.

Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$). What is the maximal possible number of different integers on the blackboard?

A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .

For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.

Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: \[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\] Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ \[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]

$(1)$ Find $\lim_{x \to \infty} \frac{x^2}{e^x}$ $(2)$ Find $\lim_{k \to 0} \frac{1}{k} \int_{0}^{k} (1 + \sin 2x)^{\frac{1}{x}} dx$

Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively. The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$. Shows that $KM$ cannot be paralel to $IH$. PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect. PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students. Sorry if I had some English mistakes here.

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

Given a parallelogram $ABCD$, let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$, the distance from vertex $B$ to $\mathcal{P}$ is $25$, and the distance from vertex $C$ to $\mathcal{P}$ is $36$. Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$. [i]Proposed by [b]HrishiP[/b][/i]

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\]

Circle $\omega$ inscribed in triangle $ABC$ touches sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$ and $C_1$ respectively. On the extension of segment $AA_1$, point $A$ is taken as point D such that $AD= AC_1$. Lines $DB_1$ and $DC_1$ intersect a second time circle $\omega$ at points $B_2$ and $C_2$. Prove that $B_2C_2$ is the diameter of circle of $\omega$.

A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.

Let $ \triangle ABC$ be a triangle with incircle $ \omega(I,r)$and circumcircle $ \zeta(O,R)$.Let $ l_{a}$ be the angle bisector of $ \angle BAC$.Denote $ P\equal{}l_{a}\cap\zeta$.Let $ D$ be the point of tangency $ \omega$ with $ [BC]$.Denote $ Q\equal{}PD\cap\zeta$.Show that $ PI\equal{}QI$ if $ PD\equal{}r$.

$2022$ lily pads are arranged in a circle. Each lily pad starts with height $1$. A frog starts on one of the lily pads, and jumps around clockwise as follows: if the frog is on a lily bad of height $k$, the lily pad grows by $1$ (becoming $k+1$), and then the frog jumps $k$ lily pads clockwise (i.e. jumping over $(k-1)$). The frog continues doing this as long as it pleases. After $n$ jumps, let $D(n)$ be the difference between the tallest lily pad and the shortest lily pad. Find, with proof, the maximum possible value of $D(n)$, or prove that $D(n)$ is unbounded.

Tom's Hat Shoppe increased all original prices by $25\%$. Now the shoppe is having a sale where all prices are $20\%$ off these increased prices. Which statement best describes the sale price of an item? $ \text{(A)}\ \text{The sale price is }5\%\text{ higher than the original price.} $ $ \text{(B)}\ \text{The sale price is higher than the original price, but by less than }5\% . $ $ \text{(C)}\ \text{The sale price is higher than the original price, but by more than }5\% . $ $ \text{(D)}\ \text{The sale price is lower than the original price.} $ $ \text{(E)}\ \text{The sale price is the same as the original price.} $

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.