Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

Find all real solutions to the following system of equations: \[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]

Solve the system in reals: $(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=2022$ and $x+y=\frac{2021}{\sqrt{2022}}$

Pablo wrote $5$ numbers on one sheet and then wrote the numbers $6,7,8,8,9,9,10,10,11$ and $ 12$ on another sheet that he gave Sofia, indicating that those numbers are the possible sums of two of the numbers that he had hidden. Decide if with this information Sofia can determine the five numbers Pablo wrote .

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Given that $a + 19b = 3$ and $a + 1019b = 5$, what is $a + 2019b$? [b]p2.[/b] How many multiples of $3$ are there between $2019$ and $2119$, inclusive? [b]p3.[/b] What is the maximum number of quadrilaterals a $12$-sided regular polygon can be quadrangulated into? Here quadrangulate means to cut the polygon along lines from vertex to vertex, none of which intersect inside the polygon, to form pieces which all have exactly $4$ sides. [b]p4.[/b] What is the value of $|2\pi - 7| + |2\pi - 6|$, rounded to the nearest hundredth? [b]p5.[/b] In the town of EMCCxeter, there is a $30\%$ chance that it will snow on Saturday, and independently, a $40\%$ chance that it will snow on Sunday. What is the probability that it snows exactly once that weekend, as a percentage? [b]p6.[/b] Define $n!$ to be the product of all integers between $1$ and $n$ inclusive. Compute $\frac{2019!}{2017!} \times \frac{2016!}{2018!}$ . [b]p7.[/b] There are $2019$ people standing in a row, and they are given positions $1$, $2$, $3$, $...$, $2019$ from left to right. Next, everyone in an odd position simultaneously leaves the row, and the remaining people are assigned new positions from $1$ to $1009$, again from left to right. This process is then repeated until one person remains. What was this person's original position? [b]p8.[/b] The product $1234\times 4321$ contains exactly one digit not in the set $\{1, 2, 3, 4\}$. What is this digit? [b]p9.[/b] A quadrilateral with positive area has four integer side lengths, with shortest side $1$ and longest side $9$. How many possible perimeters can this quadrilateral have? [b]p10.[/b] Define $s(n)$ to be the sum of the digits of $n$ when expressed in base $10$, and let $\gamma (n)$ be the sum of $s(d)$ over all natural number divisors $d$ of $n$. For instance, $n = 11$ has two divisors, $1$ and $11$, so $\gamma (11) = s(1) + s(11) = 1 + (1 + 1) = 3$. Find the value of $\gamma (2019)$. [b]p11.[/b] How many five-digit positive integers are divisible by $9$ and have $3$ as the tens digit? [b]p12.[/b] Adam owns a large rectangular block of cheese, that has a square base of side length $15$ inches, and a height of $4$ inches. He wants to remove a cylindrical cheese chunk of height $4$, by making a circular hole that goes through the top and bottom faces, but he wants the surface area of the leftover cheese block to be the same as before. What should the diameter of his hole be, in inches? [i]Αddendum on 1/26/19: the hole must have non-zero diameter. [/i] [b]p13.[/b] Find the smallest prime that does not divide $20! + 19! + 2019!$. [b]p14.[/b] Convex pentagon $ABCDE$ has angles $\angle ABC = \angle BCD = \angle DEA = \angle EAB$ and angle $\angle CDE = 60^o$. Given that $BC = 3$, $CD = 4$, and $DE = 5$, find $EA$. [i]Addendum on 1/26/19: ABCDE is specified to be convex. [/i] [b]p15.[/b] Sophia has $3$ pairs of red socks, $4$ pairs of blue socks, and $5$ pairs of green socks. She picks out two individual socks at random: what is the probability she gets a pair with matching color? [b]p16.[/b] How many real roots does the function $f(x) = 2019^x - 2019x - 2019$ have? [b]p17.[/b] A $30-60-90$ triangle is placed on a coordinate plane with its short leg of length $6$ along the $x$-axis, and its long leg along the $y$-axis. It is then rotated $90$ degrees counterclockwise, so that the short leg now lies along the $y$-axis and long leg along the $x$-axis. What is the total area swept out by the triangle during this rotation? [b]p18.[/b] Find the number of ways to color the unit cells of a $2\times 4$ grid in four colors such that all four colors are used and every cell shares an edge with another cell of the same color. [b]p19.[/b] Triangle $\vartriangle ABC$ has centroid $G$, and $X, Y,Z$ are the centroids of triangles $\vartriangle BCG$, $\vartriangle ACG$, and $\vartriangle ABG$, respectively. Furthermore, for some points $D,E, F$, vertices $A,B,C$ are themselves the centroids of triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, respectively. If the area of $\vartriangle XY Z = 7$, what is the area of $\vartriangle DEF$? [b]p20.[/b] Fhomas orders three $2$-gallon jugs of milk from EMCCBay for his breakfast omelette. However, every jug is actually shipped with a random amount of milk (not necessarily an integer), uniformly distributed between $0$ and $2$ gallons. If Fhomas needs $2$ gallons of milk for his breakfast omelette, what is the probability he will receive enough milk? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f :[0, +\infty) \rightarrow [0, +\infty)$ for which $f(f(x)+f(y)) = xy f (x+y)$ for every two non-negative real numbers $x$ and $y$.

Let $f(x)$ be a linear function and $k,l,m$ - pairwise different real numbers. It is known that $f(k)=l^3+m^3$, $f(l)=m^3+k^3$ and $f(m)=k^3+l^3$. Find the value of $k+l+m$.

Let $a, b$ and $c$ be real numbers such that $0 < a, b, c < 1$. Prove that: $$\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.$$

Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value of $a+b+c+d+e$?

Given a positive integer $n \geq 2$. Solve the following system of equations: $ \begin{cases} \ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| \\ \ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| \\ &\dots \\ \ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. \\ \end{cases} $

Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation: \[ \sqrt{x_1\minus{}1}\plus{}2\cdot \sqrt{x_2\minus{}4}\plus{}3\cdot \sqrt{x_3\minus{}9}\plus{}\cdots\plus{}n\cdot\sqrt{x_n\minus{}n^2}\equal{}\frac{1}{2}(x_1\plus{}x_2\plus{}x_3\plus{}\cdots\plus{}x_n).\]

[b]p1.[/b] Find a real number $x$ for which $x\lfloor x \rfloor = 1234.$ Note: $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. [b]p2.[/b] Let $C_1$ be a circle of radius $1$, and $C_2$ be a circle that lies completely inside or on the boundary of $C_1$. Suppose$ P$ is a point that lies inside or on $C_2$. Suppose $O_1$, and $O_2$ are the centers of $C_1$, and $C_2$, respectively. What is the maximum possible area of $\vartriangle O_1O_2P$? Prove your answer. [b]p3.[/b] The numbers $1, 2, . . . , 99$ are written on a blackboard. We are allowed to erase any two distinct (but perhaps equal) numbers and replace them by their nonnegative difference. This operation is performed until a single number $k$ remains on the blackboard. What are all the possible values of $k$? Prove your answer. Note: As an example if we start from $1, 2, 3, 4$ on the board, we can proceed by erasing $1$ and $2$ and replacing them by $1$. At that point we are left with $1, 3, 4$. We may then erase $3$ and $4$ and replacethem by $1$. The last step would be to erase $1$, $1$ and end up with a single $0$ on the board. [b]p4.[/b] Let $a, b$ be two real numbers so that $a^3 - 6a^2 + 13a = 1$ and $b^3 - 6b^2 + 13b = 19$. Find $a + b$. Prove your answer. [b]p5.[/b] Let $m, n, k$ be three positive integers with $n \ge k$. Suppose $A =\prod_{1\le i\le j\le m} gcd(n + i, k + j) $ is the product of $gcd(n + i, k + j)$, where $i, j$ range over all integers satisfying $1\le i\le j\le m$. Prove that the following fraction is an integer $$\frac{A}{(k + 1) \dots(k + m)}{n \choose k}.$$ Note: $gcd(a, b)$ is the greatest common divisor of $a$ and $b$, and ${n \choose k}= \frac{n!}{k!(n - k)!}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

Determine all integer solutions (a,b,c) with $c\leq 94$ for which: $(a+\sqrt c)^2+(b+\sqrt c)^2 = 60 + 20\sqrt c$

Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0=1$ and $x_{n+1}=\sin(x_n)+\frac{\pi} {2}-1$ for all $n \geq 0$. Show that the sequence converges and find its limit.

What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\] $ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $

Let be three nonzero rational numbers $ a,b,c $ under the relation $ (a+b)(b+c)(c+a)=a^2b^2c^2. $ Show that the expression $ \sqrt[3]{3+1/a^3+1/b^3+1/c^3} $ is rational. [i]Ion Bursuc[/i]

a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$ b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$

Suppose Bob begins walking at a constant speed from point $N$ to point $S$ along the path indicated by the following figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f5819267020f2bd38e52c6e873a2cf91ce8c49.png[/img] After Bob has walked a distance of $x$, Alice begins walking at point $N$, heading towards point $S$ along the same path. Alice walks $1.28$ times as fast as Bob when they are on the same line segment and $1.06$ times as fast as Bob otherwise. For what value of $x$ do Alice and Bob meet at point $S$?

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.

Let $a$ and $b$ be positive real numbers with $a + b =1$. Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$