A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.

The rails on a railroad are $ 30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in miles per hour is approximately the number of clicks heard in: $ \textbf{(A)}\ 20\text{ seconds} \qquad\textbf{(B)}\ 2\text{ minutes} \qquad\textbf{(C)}\ 1\frac{1}{2}\text{ minutes} \qquad\textbf{(D)}\ 5\text{ minutes}\\ \textbf{(E)}\ \text{none of these}$

Petya and Vasya are playing a game on an initially empty \(100 \times 100\) grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?

It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$. Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$. Compute $Q(-2)$.

$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?

It is known that $ \angle BAC$ is the smallest angle in the triangle $ ABC$. The points $ B$ and $ C$ divide the circumcircle of the triangle into two arcs. Let $ U$ be an interior point of the arc between $ B$ and $ C$ which does not contain $ A$. The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AU$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$. Show that $ AU \equal{} TB \plus{} TC$. [i]Alternative formulation:[/i] Four different points $ A,B,C,D$ are chosen on a circle $ \Gamma$ such that the triangle $ BCD$ is not right-angled. Prove that: (a) The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AD$ at certain points $ W$ and $ V,$ respectively, and that the lines $ CV$ and $ BW$ meet at a certain point $ T.$ (b) The length of one of the line segments $ AD, BT,$ and $ CT$ is the sum of the lengths of the other two.

[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A four-digit number $YEAR$ is called [i]very good[/i] if the system \begin{align*} Yx+Ey+Az+Rw& =Y\\ Rx+Yy+Ez+Aw & = E\\\ Ax+Ry+Yz+Ew & = A\\ Ex+Ay+Rz+Yw &= R \end{align*} of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$s in the 21st century. (The $21$st century starts in $2001$ and ends in $2100$.) [i]Proposed by Tomáš Bárta, Charles University, Prague[/i]

Let $ ABC $ be a triangle, let $ O $ and $ \gamma $ be its circumcentre and circumcircle, respectively, and let $ P $ and $ Q $ be distinct points in the interior of $ \gamma $ such that $ O, P $ and $ Q $ are not collinear. Reflect $ O $ in the midpoint of the segment $ PQ $ to obtain $R,$ then reflect $R$ in the centre of the nine-point circle of the triangle $ABC$ to obtain $S.$ The circle through $P$ and $Q$ and orthogonal to $ \gamma , $ crosses the rays $OP$ and $OQ,$ emanating from $O,$ again at $P'$ and $Q'$ respectively. Let the lines $PQ'$ and $QP'$ cross at $T.$ Prove that, if $P$ and $Q$ are isogonally conjugate with respect to the triangle $ABC,$ then so are $S$ and $T.$ [i]E.D. Camier[/i]

Evaluate $\lim_{n\to\infty}\left[\left(\prod_{k=1}^{n}\frac{2k}{2k-1}\right)\int_{-1}^{\infty}\frac{(\cos x)^{2n}}{2^x} \, dx\right]$.

For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$ Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$

Consider all lines which meet the graph of $$y=2x^4+7x^3+3x-5$$ in four distinct points, say $(x_i,y_i), i=1,2,3,4.$ Show that $$\frac{x_1+x_2+x_3+x_4}{4}$$ is independent of the line and find its value.

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?

Let $n$ be a positive integer. On a number line, Azer is at point $0$ in his car which have fuel capacity of $2^n$ units and is initially full. At each positive integer $m$, there is a gas station. Azer only moves to the right with constant speed and doesn't stop anywhere except the gas stations. Each time his car moves to the right by some amount, its fuel decreases by the same amount. Azer may choose to stop at a gas station or pass it. There are thieves at some gas stations. (A station may have multiple thieves) If Azer stops at a station which have $k\ge 0$ thieves while its car have fuel capacity $d$, his cars new fuel capacity becomes $\frac{d}{2^k}$. After that, Azer fulls his cars tank and leaves the station. Find the minimum number of thieves needed to guarantee that Azer will eventually run out of fuel. Proposed by[i] Mehmet Can Baştemir[/i] and [i]Deniz Can Karaçelebi[/i]

Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.

How many positive three-digit integers $\underline{a}\underline{b}\underline{c}$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\underline{b}\underline{c}$ corresponds to the day in that month, or $\underline{a}\underline{b}$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st. (Note: During the actual test they had to write the number of days in each month so don't feel bad if you have to google that :P)

Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit: \[ \lim_{n \to \infty} \frac{S(n)}{n^3}. \]

Let $ ABCD$ be a trapezoid inscribed in a circle $ \mathcal{C}$ with $ AB\parallel CD$, $ AB\equal{}2CD$. Let $ \{Q\}\equal{}AD\cap BC$ and let $ P$ be the intersection of tangents to $ \mathcal{C}$ at $ B$ and $ D$. Calculate the area of the quadrilateral $ ABPQ$ in terms of the area of the triangle $ PDQ$.

Find, as a function of $\, n, \,$ the sum of the digits of \[ 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), \] where each factor has twice as many digits as the previous one.

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$) such that: \[ \text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}\]

We call a set of $6$ points in the plane [i]splittable[/i] if we if can denote its elements by $A,B,C,D,E$ and $F$ in such a way that $\triangle ABC$ and $\triangle DEF$ have the same centroid. [list=a] [*]Construct a splittable set. [*]Show that any set of $7$ points has a subset of $6$ points which is [i]not[/i] splittable. [/list]