Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Consider a regular $n$-gon with $n>3$, call a line [i]acceptable[/i] if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons. (a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having $3$ or $4$ sides. (b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have $3$ or $4$ sides.

How many factors of $(20^{12})^2$ less than $20^{12}$ are not factors of $20^{12}$ ?

In a $4\times 4$ grid of unit squares, five squares are chosen at random. The probability that no two chosen squares share a side is $\tfrac mn$ for positive relatively prime integers $m$ and $n$. Find $m+n$. [i]Proposed by David Altizio[/i]

As shown in the figure, $BQ$ is a diameter of the circumcircle of $ABC$, and $D$ is the midpoint of arc $BC$ (excluding point $A$) . The bisector of the exterior angle of $\angle BAC$ intersects and the extension of $BC$ at point $E$. The ray $EQ$ intersects $\odot (ABC)$ at point $P$. Point $S$ lies on $PQ$ so that $SA = SP$. Point $T$ lies on $BC$ such that $TB = TD$. Prove that $TS \perp SE$. [img]https://cdn.artofproblemsolving.com/attachments/c/4/01460565e70b32b29cddb65d92e041bea40b25.png[/img]

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that (i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$ (ii) $ f(0)=1,$ and (iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite. Prove that $f$ is unique, and express $f(x)$ in closed form.

[b]p1.[/b] A very large lucky number $N$ consists of eighty-eight $8$s in a row. Find the remainder when this number $N$ is divided by $6$. [b]p2.[/b] If $3$ chickens can lay $9$ eggs in $4$ days, how many chickens does it take to lay $180$ eggs in $ 8$ days? [b]p3.[/b] Find the ordered pair $(x, y)$ of real numbers satisfying the conditions $x > y$, $x+y = 10$, and $xy = -119$. [b]p4.[/b] There is pair of similar triangles. One triangle has side lengths $4, 6$, and $9$. The other triangle has side lengths $ 8$, $12$ and $x$. Find the sum of two possible values of $x$. [b]p5.[/b] If $x^2 +\frac{1}{x^2} = 3$, there are two possible values of $x +\frac{1}{x}$. What is the smaller of the two values? [b]p6.[/b] Three flavors (chocolate strawberry, vanilla) of ice cream are sold at Brian’s ice cream shop. Brian’s friend Zerg gets a coupon for $10$ free scoops of ice cream. If the coupon requires Zerg to choose an even number of scoops of each flavor of ice cream, how many ways can he choose his ice cream scoops? (For example, he could have $6$ scoops of vanilla and $4$ scoops of chocolate. The order in which Zerg eats the scoops does not matter.) [b]p7.[/b] David decides he wants to join the West African Drumming Ensemble, and thus he goes to the store and buys three large cylindrical drums. In order to ensure none of the drums drop on the way home, he ties a rope around all of the drums at their mid sections so that each drum is next to the other two. Suppose that each drum has a diameter of $3.5$ feet. David needs $m$ feet of rope. Given that $m = a\pi + b$, where $a$ and $b$ are rational numbers, find sum $a + b$. [b]p8.[/b] Segment $AB$ is the diameter of a semicircle of radius $24$. A beam of light is shot from a point $12\sqrt3$ from the center of the semicircle, and perpendicular to $AB$. How many times does it reflect off the semicircle before hitting $AB$ again? [b]p9.[/b] A cube is inscribed in a sphere of radius $ 8$. A smaller sphere is inscribed in the same sphere such that it is externally tangent to one face of the cube and internally tangent to the larger sphere. The maximum value of the ratio of the volume of the smaller sphere to the volume of the larger sphere can be written in the form $\frac{a-\sqrt{b}}{36}$ , where $a$ and $b$ are positive integers. Find the product $ab$. [b]p10.[/b] How many ordered pairs $(x, y)$ of integers are there such that $2xy + x + y = 52$? [b]p11.[/b] Three musketeers looted a caravan and walked off with a chest full of coins. During the night, the first musketeer divided the coins into three equal piles, with one coin left over. He threw it into the ocean and took one of the piles for himself, then went back to sleep. The second musketeer woke up an hour later. He divided the remaining coins into three equal piles, and threw out the one coin that was left over. He took one of the piles and went back to sleep. The third musketeer woke up and divided the remaining coins into three equal piles, threw out the extra coin, and took one pile for himself. The next morning, the three musketeers gathered around to divide the coins into three equal piles. Strangely enough, they had one coin left over this time as well. What is the minimum number of coins that were originally in the chest? [b]p12.[/b] The diagram shows a rectangle that has been divided into ten squares of different sizes. The smallest square is $2 \times 2$ (marked with *). What is the area of the rectangle (which looks rather like a square itself)? [img]https://cdn.artofproblemsolving.com/attachments/4/a/7b8ebc1a9e3808096539154f0107f3e23d168b.png[/img] [b]p13.[/b] Let $A = (3, 2)$, $B = (0, 1)$, and $P$ be on the line $x + y = 0$. What is the minimum possible value of $AP + BP$? [b]p14.[/b] Mr. Mustafa the number man got a $6 \times x$ rectangular chess board for his birthday. Because he was bored, he wrote the numbers $1$ to $6x$ starting in the upper left corner and moving across row by row (so the number $x + 1$ is in the $2$nd row, $1$st column). Then, he wrote the same numbers starting in the upper left corner and moving down each column (so the number $7$ appears in the $1$st row, $2$nd column). He then added up the two numbers in each of the cells and found that some of the sums were repeated. Given that $x$ is less than or equal to $100$, how many possibilities are there for $x$? [b]p15.[/b] Six congruent equilateral triangles are arranged in the plane so that every triangle shares at least one whole edge with some other triangle. Find the number of distinct arrangements. (Two arrangements are considered the same if one can be rotated and/or reflected onto another.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\ bc + d + a = 5 \\ cd + a + b = 2 \\ da + b + c = 6 \end{cases}$

Let $n\in \mathbb{Z}_{> 0}$. The set $S$ contains all positive integers written in decimal form that simultaneously satisfy the following conditions: [list=1][*] each element of $S$ has exactly $n$ digits; [*] each element of $S$ is divisible by $3$; [*] each element of $S$ has all its digits from the set $\{3,5,7,9\}$ [/list] Find $\mid S\mid$

What is the larget integer $n$ such that $5^n$ divides $\frac {2006!}{(1003!)^2}$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 500 $

Let $A_1$ be the midpoint of the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC.{}$ The point $A_1$ is reflected relative to the side $BC,$ and then its image is reflected relative to the bisector of $\angle BAC{}$ resulting in the point $A_2 $. Similarly, the points $B_2$ and $C_2$ are constructed. Prove that the circumcenter and incenter of the triangle $ABC{}$ lie on the Euler line of the triangle $A_2B_2C_2.$ [i]Proposed by A. Tereshin[/i]

The checker moves from the lower left corner of the board $100 \times 100$ to the right top corner, moving at each step one cell to the right or one cell up. Let $a$ be the number of paths in which exactly $70$ steps the checker take under the diagonal going from the lower left corner to the upper right corner, and $b$ is the number of paths in which such steps are exactly $110$. What is more: $a$ or $b$?

Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?

If the $x$-coordinate $\overline{x}$ of the center of mass of the area lying between the $x$-axis and the curve $y=f(x)$ with $f(x)>0$, and between the lines $x=0$ and $x=a$ is given by $$\overline{x}=g(a),$$ show that $$f(x)=A\cdot \frac{g'(x)}{(x-g(x))^{2}} \cdot e^{\int \frac{1}{t-g(t)} dt},$$ where $A$ is a positive constant.

Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.

Let $A$ be the locus of points $(\alpha, \beta, \gamma)$ in the $\alpha\beta\gamma$-coordinate space that satisfy the following properties: [b](I)[/b] We have $\alpha$, $\beta$, $\gamma > 0$. [b](II)[/b] We have $\alpha + \beta + \gamma = \pi$. [b](III)[/b] The intersection of the three cylinders in the $xyz$-coordinate space given by the equations \begin{eqnarray*} y^2 + z^2 & = & \sin^2 \alpha \\ z^2 + x^2 & = & \sin^2 \beta \\ x^2 + y^2 & = & \sin^2 \gamma \end{eqnarray*} is nonempty. Determine the area of $A$.

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]