Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then \[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\] For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.

Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

How many integers between $1000$ and $2000$ have all three of the numbers $15$, $20$, and $25$ as factors? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Determine the values of the positive integer $n$ for which $$A =\sqrt{\frac{9n - 1}{n + 7}}$$ is rational.

For all real numbers $a$ prove that $3(a^4+a^2+1)\geq (a^2+a+1)^2$

Prove that for any triangle the bisector lies between the median and the height drawn from the same vertex.

Anton has an isosceles right triangle, which he wants to cut into $9$ triangular parts in the way shown in the picture. What is the largest number of the resulting $9$ parts that can be equilateral triangles? A more formal description of partitioning. Let triangle $ABC$ be given. We choose two points on its sides so that they go in the order $AC_1C_2BA_1A_2CB_1B_2$, and no two coincide. In addition, the segments $C_1A_2$, $A_1B_2$ and $B_1C_2$ must intersect at one point. Then the partition is given by segments $C_1A_2$, $A_1B_2$, $B_1C_2$, $A_1C_2$, $B_1A_2$ and $C_1B_2$. [img]https://cdn.artofproblemsolving.com/attachments/0/5/5dd914b987983216342e23460954d46755d351.png[/img]

I roll three special six-sided dice. Each die has faces labeled U, S, M, C, A, or *. The star can represent any of U, S, M, C, A. What is the probability that I can arrange the dice to spell out USA? (For instance, A*U is valid, but UU* is not valid.)

In the multiplication question, the sum of the digits in the four boxes is [img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy83L2NmMTU0MzczY2FhMGZhM2FjMjMwZDcwYzhmN2ViZjdmYjM4M2RmLnBuZw==&rn=U2NyZWVuc2hvdCAyMDE3LTAyLTI1IGF0IDUuMzguMjYgUE0ucG5n[/img] $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 22$

The figure below shows a $90 \times90$ square with each side divided into three equal segments. Some of the endpoints of these segments are connected by straight lines. Find the area of the shaded region. [asy] import graph; size(6cm); real labelscalefactor = 0.5; pen dotstyle = black; draw((-4,6)--(86,6)--(86,96)--(-4,96)--cycle); filldraw((16,76)--(-4,36)--(32,60)--(56,96)--cycle,grey); filldraw((32,60)--(-4,6)--(50,42)--(86,96)--cycle,grey); filldraw((50,42)--(26,6)--(66,26)--(86,66)--cycle,grey); draw((-4,6)--(26,6)); draw((26,6)--(56,6)); draw((56,6)--(86,6)); draw((-4,6)--(86,6)); draw((86,6)--(86,96)); draw((86,96)--(-4,96)); draw((-4,96)--(-4,6)); draw((26,96)--(-4,36)); draw((56,96)--(-4,6)); draw((86,96)--(26,6)); draw((86,66)--(56,6)); draw((-4,66)--(56,96)); draw((-4,36)--(86,96)); draw((-4,6)--(86,66)); draw((26,6)--(86,36)); draw((16,76)--(-4,36)); draw((-4,36)--(32,60)); draw((32,60)--(56,96)); draw((56,96)--(16,76)); draw((32,60)--(-4,6)); draw((-4,6)--(50,42)); draw((50,42)--(86,96)); draw((86,96)--(32,60)); draw((50,42)--(26,6)); draw((26,6)--(66,26)); draw((66,26)--(86,66)); draw((86,66)--(50,42)); dot((-4,96),dotstyle); dot((26,96),dotstyle); dot((56,96),dotstyle); dot((86,96),dotstyle); dot((-4,6),dotstyle); dot((-4,36),dotstyle); dot((-4,66),dotstyle); dot((27.09,6),dotstyle); dot((56,6),dotstyle); dot((86,36),dotstyle); dot((86,66),dotstyle); dot((86,6),dotstyle); [/asy]

In the following expression, Melanie changed some of the plus signs to minus signs: $$ 1 + 3+5+7+\cdots+97+99$$ When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs? $ \textbf{(A) }14 \qquad \textbf{(B) }15 \qquad \textbf{(C) }16 \qquad \textbf{(D) }17 \qquad \textbf{(E) }18 \qquad $

Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other?

If $15$ and $9$ are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer ?

Find all polynomials $P$ with real coefficients having no repeated roots, such that for any complex number $z$, the equation $zP(z) = 1$ holds if and only if $P(z-1)P(z + 1) = 0$. Remark: Remember that the roots of a polynomial are not necessarily real numbers.

Define a sequence $a_n$ satisfying : \[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\] Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$

In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses. Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors. [i]M. Karpuk[/i]

Diameter $ \overline{AB}$ of a circle with center $ O$ is $ 10$ units. $ C$ is a point $ 4$ units from $ A$, and on $ \overline{AB}$. $ D$ is a point $ 4$ units from $ B$, and on $ \overline{AB}$. $ P$ is any point on the circle. Then the broken-line path from $ C$ to $ P$ to $ D$: $ \textbf{(A)}\ \text{has the same length for all positions of }{P}\qquad\\ \textbf{(B)}\ \text{exceeds }{10}\text{ units for all positions of }{P}\qquad \\ \textbf{(C)}\ \text{cannot exceed }{10}\text{ units}\qquad \\ \textbf{(D)}\ \text{is shortest when }{\triangle CPD}\text{ is a right triangle}\qquad \\ \textbf{(E)}\ \text{is longest when }{P}\text{ is equidistant from }{C}\text{ and }{D}.$

Can a paper circle be cut into pieces and then rearranged into a square of the same area, if only a finite number of cuts is allowed and they must be along segments of straight lines or circular arcs? (A Belov)

Find all positive integers $n$ such that there are $\infty$ many lines of Pascal's triangle that have entries coprime to $n$ only. In other words: such that there are $\infty$ many $k$ with the property that the numbers $\binom{k}{0},\binom{k}{1},\binom{k}{2},...,\binom{k}{k}$ are all coprime to $n$.

We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[/i] a multiple of $7$.

Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that $$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.

The sequence of integers $(a_{n})_{n \ge 1}$ is defined by $a_{1}= 2$, $a_{n+1}= 2a_{n}^{2}-1$. Prove that for each positive integer n, $n$ and $a_{n}$ are coprime.

Nicky has a circle. To make his circle look more interesting, he draws a regular 15-gon, 21-gon, and 35-gon such that all vertices of all three polygons lie on the circle. Let $n$ be the number of distinct vertices on the circle. Find the sum of the possible values of $n$. [i]Proposed by Yang Liu[/i]

The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude? $ \textbf{a)}\ 4 \qquad\textbf{b)}\ 7 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 12 \qquad\textbf{e)}\ 23 $

Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$. A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove, Points $B,Q,R,P$ are concyclic.