Denote by $S$ the set of all primes $p$ such that the decimal representation of $\frac{1}{p}$ has the fundamental period of divisible by $3$. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}= 0.a_{1}a_{2}\cdots a_{3r}a_{1}a_{2}\cdots a_{3r}\cdots,\] where $r=r(p)$. For every $p \in S$ and every integer $k \ge 1$ define \[f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.\] [list=a] [*] Prove that $S$ is finite. [*] Find the highest value of $f(k, p)$ for $k \ge 1$ and $p \in S$.[/list]

Evaluate \[\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)\]

Prove $ \ \ \ \frac{22}{7}\minus{}\pi \equal{}\int_0^1 \frac{x^4(1\minus{}x)^4}{1\plus{}x^2}\ dx$.

Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\ (a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\ (b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds. [i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. [img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img] If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.

Find the smallest positive integer $n$ such that $32^n = 167x + 2$ for some integer $x$

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

What is the value of $$(2^2-2) - (3^2-3) + (4^2-4)?$$ $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 12$

Consider triangle $ABC$ with orthocenter $H$. Let points $M$ and $N$ be the midpoints of segments $BC$ and $AH$. Point $D$ lies on line $MH$ so that $AD\parallel BC$ and point $K$ lies on line $AH$ so that $DNMK$ is cyclic. Points $E$ and $F$ lie on lines $AC$ and $AB$ such that $\angle EHM=\angle C$ and $\angle FHM=\angle B$. Prove that points $D,E,F$ and $K$ lie on a circle. [i]Proposed by Alireza Dadgarnia[/i]

There are 2014 houses in a circle. Let $A$ be one of these houses. Santa Claus enters house $A$ and leaves a gift. Then with probability $1/2$ he visits $A$'s left neighbor and with probability $1/2$ he visits $A$'s right neighbor. He leaves a gift also in that second house, and then repeats the procedure (visits with probability $1/2$ either of the neighbors, leaves a gift, etc). Santa finishes as soon as every house has received at least one gift. Prove that any house $B$ different from $A$ has a probability of $1/2013$ of being the last house receiving a gift.

Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.

Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ and $CC'$ are concurrent. [i]proposed by Amirhossein Zabeti[/i]

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. [i]N. Agakhanov, N. Tereshin[/i]

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

Consider the set $S = \{1, 2, . . . , 2015\}$. How many ways are there to choose $2015$ distinct (possibly empty and possibly full) subsets $X_1, X_2, . . . , X_{2015}$ of $S$ such that $X_i$ is strictly contained in $X_{i+1}$ for all $1 \le i \le 2014$?

For a randomly selected permutation $ \mathbf{f} = (f_1,..., f_n) $ of the set $ \{1,\ldots, n\} $ let us denote by $ X(\mathbf{f}) $ the largest number $ k \leq n $ such that $ f_i < f_{ i+1} $ for all numbers $ i < k $. Prove that the expected value of the random variable $ X $ is $ \sum_{k=1}^n \frac{1}{k!} $.

The value of $$\left(1+\frac26+\frac{2\cdot5}{6\cdot12}+\frac{2\cdot5\cdot8}{6\cdot12\cdot18}+\ldots\right)^3$$

Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

For reals $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$. [i]Proposed by Evan Chen[/i]

Gizmo is thinking of a geometric sequence in which the third term is $1215$ and the fifth is $540$. Which of the following could be the eighth term of Gizmo's sequence? $\text{(A) }-160\qquad\text{(B) }-135.5\qquad\text{(C) }216\qquad\text{(D) }240\qquad\text{(E) }472.5$

Let $P$ be the set of points in the Euclidean plane, and let $L$ be the set of lines in the same plane. Does there exist an one-to-one mapping (injective function) $f : L \to P$ such that for each $\ell \in L$ we have $f(\ell) \in \ell$?

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?

Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$. $AN$ and $MB$ intersect at $X$. Given that $\angle AXB$ measures $127^{\circ}$ and the minor arc $AM$ measures $14^{\circ}$, compute the measure of the angle at $P$. [asy] size(200); defaultpen(fontsize(10pt)); pair P=(40,10),C=(-20,10),K=(-20,-10); path CC=circle((0,0),20), PC=P--C, PK=P--K; pair A=intersectionpoints(CC,PC)[0], B=intersectionpoints(CC,PC)[1], M=intersectionpoints(CC,PK)[0], N=intersectionpoints(CC,PK)[1], X=intersectionpoint(A--N,B--M); draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M); label("$A$",A,plain.NE);label("$B$",B,plain.NW);label("$M$",M,SE); label("$P$",P,E);label("$N$",N,dir(250));label("$X$",X,plain.N);[/asy]

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.