Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$

Let $n$ be the number of $2024$ digit base-$10$ numbers that satisfy the property $f(9x) = x$, where $f$ is the function that reverses the base-$10$ digits of a number. Estimate the number of digits in the base-$10$ representation of $n$. \\\\ Your score will be calculated by the function $\max(0, \lfloor\frac{200\log_{10}A}{(A - S)^2+10\log_{10}A}\rfloor)$, where $S$ is your submission and $A$ is the true answer. [i]Lightning 5.3[/i]

$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.

Let $n$ and $k$ be positive integers, with $n\geq k+1$. There are $n$ countries on a planet, with some pairs of countries establishing diplomatic relations between them, such that each country has diplomatic relations with at least $k$ other countries. An evil villain wants to divide the countries, so he executes the following plan: (1) First, he selects two countries $A$ and $B$, and let them lead two allies, $\mathcal{A}$ and $\mathcal{B}$, respectively (so that $A\in \mathcal{A}$ and $B\in\mathcal{B}$). (2) Each other country individually decides wether it wants to join ally $\mathcal{A}$ or $\mathcal{B}$. (3) After all countries made their decisions, for any two countries with $X\in\mathcal{A}$ and $Y\in\mathcal{B}$, eliminate any diplomatic relations between them. Prove that, regardless of the initial diplomatic relations among the countries, the villain can always select two countries $A$ and $B$ so that, no matter how the countries choose their allies, there are at least $k$ diplomatic relations eliminated. [i]Proposed by YaWNeeT.[/i]

The point $ O$ is the center of the circle circumscribed about $ \triangle ABC$, with $ \angle BOC \equal{} 120^\circ$ and $ \angle AOB \equal{} 140^\circ$, as shown. What is the degree measure of $ \angle ABC$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair B=dir(80), A=dir(220), C=dir(320), O=(0,0); draw(unitcircle); draw(A--B--C--O--A--C); draw(O--B); draw(anglemark(C,O,A,2)); label("$A$",A,SW); label("$B$",B,NNE); label("$C$",C,SE); label("$O$",O,S); label("$140^{\circ}$",O,NW,fontsize(8pt)); label("$120^{\circ}$",O,ENE,fontsize(8pt));[/asy]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

Estimate the number of characters, excluding spaces, in the \LaTeX~source file for this Lightning Round, which includes the answer sheets and exactly one Asymptote diagram. Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{20}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate? [i]Lightning 5.3[/i]

The diagram below shows equilateral $\triangle ABC$ with side length $2$. Point $D$ lies on ray $\overrightarrow{BC}$ so that $CD = 4$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $E$, $F$, and $D$ are collinear, and the area of $\triangle AEF$ is half of the area of $\triangle ABC$. Then $\tfrac{AE}{AF}=\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + 2n$. [asy] import math; size(7cm); pen dps = fontsize(10); defaultpen(dps); dotfactor=4; pair A,B,C,D,E,F; B=origin; C=(2,0); D=(6,0); A=(1,sqrt(3)); E=(1/3,sqrt(3)/3); F=extension(A,C,E,D); draw(C--A--B--D,linewidth(1.1)); draw(E--D,linewidth(.7)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$F$",F,NE); [/asy]

Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$. [hide=original wording] Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]

Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points. Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong). (a) Does there exist a finite set of points, not all collinear, such that a line between any two points in the set passes through a third point in the set? (b) Let $ABC$ be a triangle and $P$ be a point. The [i]isogonal conjugate[/i] of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ over the $C$-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate? (c) Let $F$ be a convex figure in a plane, and let $P$ be the largest pentagon that can be inscribed in $F$. Is it necessarily true that the area of $P$ is at least $\frac{3}{4}$ the area of $F$? (d) Is it possible to cut an equilateral triangle into $2017$ pieces, and rearrange the pieces into a square? (e) Let $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D,E,F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle{BPC}$, $PE$ bisects $\angle{CPA}$, and $PF$ bisects $\angle{APB}$. Is it necessarily true that $AP+BP+CP\ge 2(PD+PE+PF)$? (f) Let $P_{2018}$ be the surface area of the $2018$-dimensional unit sphere, and let $P_{2017}$ be the surface area of the $2017$-dimensional unit sphere. Is $P_{2018}>P_{2017}$? [color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color]

[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$ ($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits). [b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table: the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game. a) The player who makes the first move can win the game. What is the winning first move? b) How can he win? (Describe his strategy.) [b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover? [b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back. [b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p, q$ be real numbers with $0 < p < q$ and let $t_1, t_2, \cdots, t_n$ be real numbers in the interval $[p, q]$. Denote by $A$ and $B$ the arithmetic means of $t_1, t_2, \cdots, t_n$ and of $t_1^2, t_2^2,\cdots , t_n^2$, respectively. Prove that \[\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.\]

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

Let $ K$ be a point inside the triangle $ ABC$. Let $ M$ and $ N$ be points such that $ M$ and $ K$ are on opposite sides of the line $ AB$, and $ N$ and $ K$ are on opposite sides of the line $ BC$. Assume that $ \angle MAB \equal{} \angle MBA \equal{} \angle NBC \equal{} \angle NCB \equal{} \angle KAC \equal{} \angle KCA$. Show that $ MBNK$ is a parallelogram.

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. $AC$ and $BD$ meet at $E$. The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$. Prove that the line through the incenters of $\triangle ABE\,,\, \triangle ABF$ and the line through the incenters of $\triangle CDE\,,\, \triangle CDF$ meet at a point lying on the circle $k$. [i]Proposed by N. Beluhov[/i]

Regular hexagon $P_1P_2P_3P_4P_5P_6$ has side length $2$. For $1 \le i \le 6$, let $C_i$ be a unit circle centered at $P_i$ and $\ell_i$ be one of the internal common tangents of $C_i$ and $C_{i+2}$, where $C_7 = C_1$ and $C_8 = C_2$. Assume that the lines $\{\ell_1, \ell_2, \ell_3, \ell_4, \ell_5,\ell_6\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Let $D$ be a point on the side $[AB]$ of the isosceles triangle $ABC$ such that $|AB|=|AC|$. The parallel line to $BC$ passing through $D$ intersects $AC$ at $E$. If $m(\widehat A) = 20^\circ$, $|DE|=1$, $|BC|=a$, and $|BE|=a+1$, then which of the followings is equal to $|AB|$? $ \textbf{(A)}\ 2a \qquad\textbf{(B)}\ a^2-a \qquad\textbf{(C)}\ a^2+1 \qquad\textbf{(D)}\ (a+1)^2 \qquad\textbf{(E)}\ a^2+a $

$x_1,x_2$ are two real roots of the equation $x^2-(k-2)x+(k^2+3k+5)=0$.What's the maximum value of $x_1^2+x_2^2$? $\text{(A)}19\qquad\text{(B)}18\qquad\text{(C)}5\frac{5}{9}\qquad\text{(D)}$Not exist

Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$

A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or lime with equal probability and gives the grid to her friend Sophia. Sophia then plays with the grid of blocks. She can take the top row of blocks and move it to the bottom, as shown. 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 1 2 3 Grid A Grid A0 She can also take the leftmost column of blocks and move it to the right end, as shown. 1 2 3 4 5 6 7 8 9 2 3 1 5 6 4 8 9 7 Grid B Grid B0 Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the moves described above to obtain another grid with the same coloring but a different numbering scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the top row of blocks to the bottom results in a grid with a different numbering but the same coloring as Grid A. What is the probability that Sophia receives a citrus grid of blocks?

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

The ruler of Laputa will set up a train network between cities in the state, which satisfies the following conditions: - [i]Uniformity[/i]: From one city to another, by train, possibly through exchanges. - [i]Prohibition N[/i]: There exist no four cities $A, B, C, D$ such that there are direct routes between $A$ and $B, B$ and $C$, and $C$ and $D$, but taking a shortcut is not possible, that is, there are no direct rout between $A$ and $C, B$ and $D$, or $A$ and $D$. In addition, a direct airliner connection will be established exactly between their city pairs, with no direct train connection. Prove that the airline network is not connected when there is more than one city.

Let $P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)$ be points on the plane such that $P_0P_1P_2\Delta$ contains the origin $O$. Show that the areas of triangles $P_0OP_1,P_0OP_2,P_1OP_2$ form a geometric sequence in that order if and only if there exists a real number $x$, such that $$ a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0 $$

Let $n>1$ be an integer. Given a simple graph $G$ on $n$ vertices $v_1, v_2, \dots, v_n$ we let $k(G)$ be the minimal value of $k$ for which there exist $n$ $k$-dimensional rectangular boxes $R_1, R_2, \dots, R_n$ in a $k$-dimensional coordinate system with edges parallel to the axes, so that for each $1\leq i<j\leq n$, $R_i$ and $R_j$ intersect if and only if there is an edge between $v_i$ and $v_j$ in $G$. Define $M$ to be the maximal value of $k(G)$ over all graphs on $n$ vertices. Calculate $M$ as a function of $n$.

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$