Compute the value of \[\sum_{i=0}^{2026} \frac{i^2}{9+i^4} \pmod{2027},\] where $\frac{1}{a}$ denotes the multiplicative inverse of $a$ modulo $2027$.

Let $P_1$ be a regular $n$-gon, where $n\in\mathbb{N}$. We construct $P_2$ as the regular $n$-gon whose vertices are the midpoints of the edges of $P_1$. Continuing analogously, we obtain regular $n$-gons $P_3,P_4,\ldots ,P_m$. For $m\ge n^2-n+1$, find the maximum number $k$ such that for any colouring of vertices of $P_1,\ldots ,P_m$ in $k$ colours there exists an isosceles trapezium $ABCD$ whose vertices $A,B,C,D$ have the same colour. [i]Radu Ignat[/i]

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles

Let $C_1$ and $C_2$ be two circumferences externally tangents at $S$ such that the radius of $C_2$ is the triple of the radius of $C_1$. Let a line be tangent to $C_1$ at $P \neq S$ and to $C_2$ at $Q \neq S$. Let $T$ be a point on $C_2$ such that $QT$ is diameter of $C_2$. Let the angle bisector of $\angle SQT$ meet $ST$ at $R$. Prove that $QR=RT$

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list]

Aidan owns a plot of land that is in the shape of a triangle with side lengths $5$,$10$, and $5\sqrt3$ feet. Aidan wants to plant radishes such that there are no two radishes that are less than $1$ foot apart. Determine the maximum number of radishes Aidan can plant

A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds? $\text{(A) }148\qquad\text{(B) }154\qquad\text{(C) }167\qquad\text{(D) }217\qquad\text{(E) }223$

The last digit of number $2017^{2017} - 2013^{2015}$ is (A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.

The center of a circle $\omega_2$ lies on a circle $\omega_1$. Tangents $XP$ and $XQ$ to $\omega_2$ from an arbitrary point $X$ of $\omega_1$ ($P$ and $Q$ are the touching points) meet $\omega_1$ for the second time at points $R$ and $S$. Prove that the line $PQ$ bisects the segment $RS$.

Let $a, b, c$ be positive numbers such that $abc \le 8$. Prove that $$\frac{1}{a^2 - a + 1} +\frac{1}{b^2 - b + 1}++\frac{1}{c^2 - c + 1} \ge 1$$

If the value of the infinite sum $$\frac{1}{2^2-1^2}+\frac{1}{4^2-2^2}+\frac{1}{8^2-4^2}+\frac{1}{16^2-8^2}+\dots.$$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]

For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

Find all polynomials with real coefficients, for which the equality \[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\] holds for any real number $ x$.

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$

Calculate \[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]

$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.

Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

p1. It is known that $f$ is a function such that $f(x)+2f\left(\frac{1}{x}\right)=3x$ for every $x\ne 0$. Find the value of $x$ that satisfies $f(x) = f(-x)$. p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point $O$. Point $P$ lies on side $BC$ so that $AP$ is the altitude of triangle ABC. If $\angle ABC + 30^o \le \angle ACB$, prove that $\angle COP + \angle CAB < 90^o$. p3. Find all natural numbers $a, b$, and $c$ that are greater than $1$ and different, and fulfills the property that $abc$ divides evenly $bc + ac + ab + 2$. p4. Let $A, B$, and $ P$ be the nails planted on the board $ABP$ . The length of $AP = a$ units and $BP = b$ units. The board $ABP$ is placed on the paths $x_1x_2$ and $y_1y_2$ so that $A$ only moves freely along path $x_1x_2$ and only moves freely along the path $y_1y_2$ as in following image. Let $x$ be the distance from point $P$ to the path $y_1y_2$ and y is with respect to the path $x_1x_2$ . Show that the equation for the path of the point $P$ is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. [img]https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png[/img] p5. There are three boxes $A, B$, and $C$ each containing $3$ colored white balls and $2$ red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box $A$. 2. Step 2 $\bullet$ If the ball drawn from box $A$ in step 1 is white, then the ball is put into box $B$. Next from box $B$ one ball is drawn, if it is a white ball, then the ball is put into box $C$, whereas if the one drawn is red ball, then the ball is put in box $A$. $\bullet$ If the ball drawn from box $A$ in step 1 is red, then the ball is put into box $C$. Next from box $C$ one ball is taken. If what is drawn is a white ball then the ball is put into box $A$, whereas if the ball drawn is red, the ball is placed in box $B$. 3. Step 3 Take one ball each from squares $A, B$, and $C$. What is the probability that all the balls drawn in step 3 are colored red?

Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$.

How many ten-digit whole numbers satisfy the following property: they have only $2$ and $5$ as digits, and there are no consecutive $2$'s in the number (i.e. any two $2$'s are separated by at least one $5$)?