Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if \[ \frac{n}{810}=0.d25d25d25\ldots \]

Find a positive integer $n$ such that there is a polygon with $n$ sides where each of its interior angles measures $177^o$

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$. (a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$ (b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare? [asy] unitsize(36); fill((0,0)--(1,0)--(1,1)--cycle,gray); fill((1,1)--(1,2)--(2,2)--cycle,gray); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,0)--(2,2)); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,gray); draw((3,0)--(5,0)--(5,2)--(3,2)--cycle); draw((4,0)--(4,2)); draw((3,1)--(5,1)); fill((6,1)--(6.5,0.5)--(7,1)--(7.5,0.5)--(8,1)--(7.5,1.5)--(7,1)--(6.5,1.5)--cycle,gray); draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((6,0)--(8,2)); draw((6,2)--(8,0)); draw((7,0)--(6,1)--(7,2)--(8,1)--cycle); label("$I$",(1,2),N); label("$II$",(4,2),N); label("$III$",(7,2),N); [/asy] $\text{(A)}\ \text{The shaded areas in all three are equal.}$ $\text{(B)}\ \text{Only the shaded areas of }I\text{ and }II\text{ are equal.}$ $\text{(C)}\ \text{Only the shaded areas of }I\text{ and }III\text{ are equal.}$ $\text{(D)}\ \text{Only the shaded areas of }II\text{ and }III\text{ are equal.}$ $\text{(E)}\ \text{The shaded areas of }I, II\text{ and }III\text{ are all different.}$

Jimmy’s garden has right angled triangle shape that lies on island of circular shape in such a way that the corners of the triangle are on the shore of the island. When he made fences along the garden he realized that the length of the shortest side is $36$ meter shorter than the longest side, and third side required $48$ meter long fence. In the middle of the garden he built a house of circular shape that has the largest possible size. Jimmy measured the distance between the center of his house and the center of the island. What is the square of this distance?

If real numbers $a, b, c, d, e$ satisfy $a + 1 = b + 2 = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3$, what is the value of $a^2 + b^2 + c^2 + d^2 + e^2$ ?

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

Find the sum of the answers to all even numbered Short Answer problems, with the exception of #26, rounded to the nearest tenth. [i](.7 points)[/i]

Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.

the edges of triangle $ABC$ are $a,b,c$ respectively,$b<c$,$AD$ is the bisector of $\angle A$,point $D$ is on segment $BC$. (1)find the property $\angle A$,$\angle B$,$\angle C$ have,so that there exists point $E,F$ on $AB,AC$ satisfy $BE=CF$,and $\angle NDE=\angle CDF$ (2)when such $E,F$ exist,express $BE$ with $a,b,c$

Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,

Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

All russian olympiad 2016,Day 2 ,grade 9,P8 : Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\le\frac{1}{a^2b^2c^2d^2}$$ All russian olympiad 2016,Day 2,grade 11,P7 : Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that $$\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\le\frac{1}{a^3b^3c^3d^3}$$ Russia national 2016

Let a board game has $10$ cards: $3$ [b]skull[/b] cards, $5$ [b]coin[/b] cards and $2$ [b]blank[/b] cards. We put these $10$ cards downward and shuffle them and take cards one by one from the top. Once $3$ [b]skull[/b] cards or [b]coin[/b] cards appears we stop. What is the possibility of it stops because there appears $3$ [b]skull[/b] cards?

Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.

Find all polinomials $ P(x)$ with real coefficients, such that $ P(\sqrt {3}(a \minus{} b)) \plus{} P(\sqrt {3}(b \minus{} c)) \plus{} P(\sqrt {3}(c \minus{} a)) \equal{} P(2a \minus{} b \minus{} c) \plus{} P( \minus{} a \plus{} 2b \minus{} c) \plus{} P( \minus{} a \minus{} b \plus{} 2c)$ for any $ a$,$ b$ and $ c$ real numbers

A ship travels from point A to point B along a semicircular path, centered at Island X. Then it travels along a straight path from B to C. Which of these graphs best shows the ship's distance from Island X as it moves along its course? [asy]size(150); pair X=origin, A=(-5,0), B=(5,0), C=(0,5); draw(Arc(X, 5, 180, 360)^^B--C); dot(X); label("$X$", X, NE); label("$C$", C, N); label("$B$", B, E); label("$A$", A, W);[/asy] $\textbf{(A)}$ [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw(Arc((4,10), 4, 0, 180)^^(8,10)--(16,12)); [/asy] $\textbf{(B)}$ [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw(Arc((12,10), 4, 180, 360)^^(0,10)--(8,10)); [/asy] $\textbf{(C)}$ [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw((0,8)--(10,10)--(16,8)); [/asy] $\textbf{(D)}$ [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw(Arc((12,10), 4, 0, 180)^^(0,10)--(8,10)); [/asy] $\textbf{(E)}$ [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw((0,6)--(6,6)--(16,10)); [/asy]

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

A cube has one of its vertices and all edges connected to that vertex deleted. How many ways can the letters from the word "$AMONGUS$" be placed on the remaining vertices of the cube so that one can walk along the edges to spell out "$AMONGUS$"? Note that each vertex will have at most $1$ letter, and one vertex is deleted and not included in the walk

(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals. (b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that $$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$

$2019$ coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from $ 1$ to $99$, the second player in one move can take an even number of coins from $2$ to $100$. The player who can not make a move is lost. Who has the winning strategy in this game?

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

Niki has $15$ dollars more than twice as much money as her sister Amy. If Niki gives Amy $30$ dollars, then Niki will have hals as much money as her sister. How many dollars does Niki have?

If $S$ is a sequence of positive integers let $p(S)$ be the product of the members of $S$. Let $m(S)$ be the arithmetic mean of $p(T)$ for all non-empty subsets $T$ of $S$. Suppose that $S'$ is formed from $S$ by appending an additional positive integer. If $m(S) = 13$ and $m(S') = 49$, find $S'$.