A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.

[b]p1.[/b] Find the largest $4$ digit integer that is divisible by $2$ and $5$, but not $3$. [b]p2.[/b] The diagram below shows the eight vertices of a regular octagon of side length $2$. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure $270^o$. Compute the area of the shaded region. [center][img]https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png[/img][/center] [b]p3.[/b] Consider the numbers formed by writing full copies of $2023$ next to each other, like so: $$2023202320232023...$$ How many copies of $2023$ are next to each other in the smallest multiple of $11$ that can be written in this way? [b]p4.[/b] A positive integer $n$ with base-$10$ representation $n = a_1a_2 ...a_k$ is called [i]powerful [/i] if the digits $a_i$ are nonzero for all $1 \le i \le k$ and $$n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .$$ What is the unique four-digit positive integer that is [i]powerful[/i]? [b]p5.[/b] Six $(6)$ chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation. [b]p6.[/b] Given that the infinite sum $\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...$ is equal to $\frac{\pi^4}{90}$, compute the value of $$\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}$$ [b]p7.[/b] Triangle $ABC$ is equilateral. There are $3$ distinct points, $X$, $Y$ , $Z$ inside $\vartriangle ABC$ that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio $1 : 1 : 2$ in some order. Find the ratio of the area of $\vartriangle ABC$ to that of $\vartriangle XY Z$. [b]p8.[/b] For a fixed prime $p$, a finite non-empty set $S = \{s_1,..., s_k\}$ of integers is $p$-[i]admissible [/i] if there exists an integer $n$ for which the product $$(s_1 + n)(s_2 + n) ... (s_k + n)$$ is not divisible by $p$. For example, $\{4, 6, 8\}$ is $2$-[i]admissible[/i] since $(4+1)(6+1)(8+1) = 315$ is not divisible by $2$. Find the size of the largest subset of $\{1, 2,... , 360\}$ that is two-,three-, and five-[i]admissible[/i]. [b]p9.[/b] Kwu keeps score while repeatedly rolling a fair $6$-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a $1$, he stops rolling. For example, if the first roll is $1$, he gets a score of $1$, and if he rolls the sequence $(3, 4, 1)$, he gets a score of $3 + 3 \cdot 4 + 2 \cdot 1 = 17$. What is his expected score? [b]p10.[/b] Let $\{a_1, a_2, a_3, ...\}$ be a geometric sequence with $a_1 = 4$ and $a_{2023} = \frac14$ . Let $f(x) = \frac{1}{7(1+x^2)}$. Find $$f(a_1) + f(a_2) + ... + f(a_{2023}).$$ [b]p11.[/b] Let $S$ be the set of quadratics $x^2 + ax + b$, with $a$ and $b$ real, that are factors of $x^{14} - 1$. Let $f(x)$ be the sum of the quadratics in $S$. Find $f(11)$. [b]p12.[/b] Find the largest integer $0 < n < 100$ such that $n^2 + 2n$ divides $4(n- 1)! + n + 4$. [b]p13.[/b] Let $\omega$ be a unit circle with center $O$ and radius $OQ$. Suppose $P$ is a point on the radius $OQ$ distinct from $Q$ such that there exists a unique chord of $\omega$ through $P$ whose midpoint when rotated $120^o$ counterclockwise about $Q$ lies on $\omega$. Find $OP$. [b]p14.[/b] A sequence of real numbers $\{a_i\}$ satisfies $$n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n$$ for each integer $n \ge 1$. Find the value of $a_{2023}$. [b]p15.[/b] In $\vartriangle ABC$, let $\angle ABC = 90^o$ and let $I$ be its incenter. Let line $BI$ intersect $AC$ at point $D$, and let line $CI$ intersect $AB$ at point $E$. If $ID = IE = 1$, find $BI$. [b]p16.[/b] For a positive integer $n$, let $S_n$ be the set of permutations of the first $n$ positive integers. If $p = (a_1, ..., a_n) \in S_n$, then define the bijective function $\sigma_p : \{1,..., n\} \to \{1, ..., n\}$ such that $\sigma_p (i) = a_i$ for all integers $1 \le i \le n$. For any two permutations $p, q \in S_n$, we say $p$ and $q$ are friends if there exists a third permutation $r \in S_n$ such that for all integers $1 \le i \le n$, $$\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).$$ Find the number of friends, including itself, that the permutation $(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)$ has in $S_{10}$. PS. You had better use hide for answers.

$-15+9\times (6\div 3) =$ $\text{(A)}\ -48 \qquad \text{(B)}\ -12 \qquad \text{(C)}\ -3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 12$

Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$.

Triangle $ABC$ has orthocentre $H$. The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$. (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)

Consider a triangle $ABC$ with height $AH$ and $H$ on $BC$. Let $\gamma_1$ and $\gamma_2$ be the circles with diameter $BH,CH$ respectively, and let their centers be $O_1$ and $O_2$. Points $X,Y$ lie on $\gamma_1,\gamma_2$ respectively such that $AX,AY$ are tangent to each circle and $X,Y,H$ are all distinct. $P$ is a point such that $PO_1$ is perpendicular to $BX$ and $PO_2$ is perpendicular to $CY$. Prove that the circumcircles of $PXY$ and $AO_1O_2$ are tangent to each other.

Calculate the sum: $nx+(n-1)x^2+...+2x^{n-1}+x^n$

Jim and Jane divide a triangular cake between themselves. Jim chooses any point in the cake and Jane makes a straight cut through this point and chooses the piece. Find the size of the piece that each of them can guarantee for himself/herself (both of them want to get as much as possible). [i](4 points)[/i]

Let $x_0(t)=1$, $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$; $y_{n,0}(t)=1$, $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$, $1\leq k \leq n$. Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$.

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$. [img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

In the figure, the area of square WXYZ is $25 \text{cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\Delta ABC$, $AB = AC$, and when $\Delta ABC$ is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of $\Delta ABC$, in square centimeters? [asy] defaultpen(fontsize(8)); size(225); pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5); draw((-4,0)--Y--X--(-4,10)--cycle); draw((0,-2)--(0,12)--(-2,12)--(-2,8)--B--A--C--(-2,2)--(-2,-2)--cycle); dot(O); label("$A$", A, NW); label("$O$", O, NE); label("$B$", B, SW); label("$C$", C, NW); label("$W$",W , NE); label("$X$", X, N); label("$Y$", Y, N); label("$Z$", Z, SE); [/asy] $ \textbf{(A)}\ \frac{15}4\qquad\textbf{(B)}\ \frac{21}4\qquad\textbf{(C)}\ \frac{27}4\qquad\textbf{(D)}\ \frac{21}2\qquad\textbf{(E)}\ \frac{27}2$

Which of the numbers $1, 2, \ldots , 1983$ has the largest number of divisors?

For every odd number $p>1$ we have: $\textbf{(A)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-2\qquad \textbf{(B)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p\\ \textbf{(C)}\ (p-1)^{\frac{1}{2}(p-1)} \; \text{is divisible by} \; p\qquad \textbf{(D)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p+1\\ \textbf{(E)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-1$

Find the $ \mathcal{C}^1 $ class functions $ f:[0,1]\longrightarrow\mathbb{R} $ satisfying the following three clauses: $ \text{(i) } f(0)=0 $ $ \text{(ii) } \text{Im} f'\subset (0,1] $ $ \text{(iii) }F(1)-\frac{\left( f(1) \right)^3}{3} =F(0)=0, $ where $ F $ is a primitive of $ f. $

Let $g(n)$ be the greatest common divisor of $n$ and $2015$. Find the number of triples $(a,b,c)$ which satisfies the following two conditions: $1)$ $a,b,c \in$ {$1,2,...,2015$}; $2)$ $g(a),g(b),g(c),g(a+b),g(b+c),g(c+a),g(a+b+c)$ are pairwise distinct.

Let $ABC$ and $A'B'C'$ be two triangles so that the midpoints of $\overline{AA'}, \overline{BB'}, \overline{CC'}$ form a triangle as well. Suppose that for any point $X$ on the circumcircle of $ABC$, there exists exactly one point $X'$ on the circumcircle of $A'B'C'$ so that the midpoints of $\overline{AA'}, \overline{BB'}, \overline{CC'}$ and $\overline{XX'}$ are concyclic. Show that $ABC$ is similar to $A'B'C'$. [i]Proposed by usjl[/i]

Which fraction is bigger: $\frac{5553}{5557}$ or $\frac{6664}{6669}$ ?

Let $P(n)$ be the number of ways to split a natural number $n$ to the sum of powers of two, when the order does not matter. For example $P(5) = 4$, as $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1$. Prove that for any natural the identity $P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}} P(1) + (-1)^{a_n} = 0,$ is true, where $a_k$ is the number of units in the binary number record $k$ . [url=http://matol.kz/comments/2720/show]source[/url]

Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$, $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$.)

Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $AD$ be the bisector of angle $A$ with $D$ on $BC$ . Let the circumcircle of triangle $ACD$ intersect $AB$ again at $E$; and let the circumcircle of triangle $ABD$ intersect $AC$ again at $F$ . Let $K$ be the reflection of $E$ in the line $BC$ . Prove that $FK = BC$.

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7)); draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

In the quadrilateral $ABCD$, we have $\angle C$ is triple of $\angle A$, let $P$ be a point in the side $AB$ such that $\angle DPA = 90º$ and let $Q$ be a point in the segment $DA$ where $\angle BQA = 90º$ the segments $DP$ and $CQ$ intersects in $O$ such that $BO = CO = DO$, find $\angle A$ and $\angle C$.

Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.