Find all functions $f : R \to R$ such that the inequality $$\sum_{i=1}^{2549} f(x_i + x_{i+1}) + f (\sum_{i=1}^{2550}x_y) \le \sum_{i=1}^{2550}f(2x_i)$$ for all reals $x_1, x_2, . . . , x_{2550}$.

Let a function $f$ satisfy $f(1) = 1$ and $f(1)+ f(2)+...+ f(n) = n^2f(n)$ for all $n \in N$. Determine $f(1995)$.

Let $a,b,c\in\mathbb R_+$ satisfy that $$\sqrt{(1+a)(1+b)(1+c)}=\sqrt{(ab-a-b+1)(1+c)}+\sqrt{(bc-b-c+1)(1+a)}+\sqrt{(ca-c-a+1)(1+b)}.$$ Find the value range of $a+b+c.$

The circle $ \omega $ inscribed in triangle $ABC$ touches its sides $AB, BC, CA$ at points $K, L, M$ respectively. In the arc $KL$ of the circle $ \omega $ that does not contain the point $M$, we select point $S$. Denote by $P, Q, R, T$ the intersection points of straight $AS$ and $KM, ML$ and $SC, LP$ and $KQ, AQ$ and $PC$ respectively. It turned out that the points $R, S$ and $M$ are collinear. Prove that the point $T$ also lies on the line $SM$.

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

Given an acute triangle $ABC$, let $P$ be an arbitrary point on segment $BC$. A line passing through $P$ and perpendicular to $AC$ intersects $AB$ at $P_b$. A line passing through $P$ and perpendicular to $AB$ intersects $AC$ at $P_c$. Prove that the circumcircle of triangle $AP_bP_c$ passes through a fixed point other than $A$ when $P$ varies on segment $BC$. [i]Proposed by ltf0501[/i]

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.

Let us have a line $\ell$ in the space and a point $A$ not lying on $\ell.$ For an arbitrary line $\ell'$ passing through $A$, $XY$ ($Y$ is on $\ell'$) is a common perpendicular to the lines $\ell$ and $\ell'.$ Find the locus of points $Y.$

In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$. [i]Proposed by Eugene Chen[/i]

Let $f : R^+ \to R^+$ be a function such that for all $x, y \in R^+$, $f(x)f(y) = f(xy) + f\left( \frac{x}{y}\right)$, where $R^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f(2^{2^{2020}})$. .

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Quadrilateral $ABCD$ has the property that $AD = BD = CD$ and $\angle ADB < \angle CDB$. Let the circumcircle of $ABD$ be $O$. $O$ intersects $BC$ at $E$ and $CD$ at $F$. Next, extend $AB$ and $CD$ to intersect at a point $G$. Suppose that $BE = 1$, $EF = 3$, and $F D = 4$. Compute the perimeter of $\vartriangle ADG$.

Given a natural number $n\ge 3$, determine all strictly increasing sequences $a_1<a_2<\cdots<a_n$ such that $\text{gcd}(a_1,a_2)=1$ and for any pair of natural numbers $(k,m)$ satisfy $n\ge m\ge 3$, $m\ge k$, $$\frac{a_1+a_2+\cdots +a_m}{a_k}$$ is a positive integer.

Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$. [i]David Yang.[/i]

It is known that the distances from all the vertices of a cube and the centers of its faces to a certain plane ($14$ values in total) take two different values. The smallest is $1$. What can the edge of a cube be equal to?

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

Consider the system of equations $$a + 2b + 3c + \ldots + 26z = 2020$$ $$b + 2c + 3d + \ldots + 26a = 2019$$ $$\vdots$$ $$y + 2z + 3a + \ldots + 26x = 1996$$ $$z + 2a + 3b + \ldots + 26y = 1995$$ where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of $$z + 2y + 3x + \dots + 26a.$$ [i]Proposed by Joshua Hsieh[/i]

Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$. If $x_1,...,x_m$ ($n\ge m\ge3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots$.

Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.

Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .

Let $$P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .$$ Calculate for all $x \in \mathbb{R},$ $$\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}$$

It is known that the real function $f(t)$ is monotonic increasing in the interval $-8 \le t \le 8$, but nothing is known about what happens outside of it. In what range of values of $x$, can it be ensured that the function $y = f(2x - x^2)$ is monotonic increasing?

You have an unlimited supply of square tiles with side length $ 1$ and equilateral triangle tiles with side length $ 1$. For which n can you use these tiles to create a convex $n$-sided polygon? The tiles must fit together without gaps and may not overlap.

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

Dagan has a wooden cube. He paints each of the six faces a different color. He then cuts up the cube to get eight identically-sized smaller cubes, each of which now has three painted faces and three unpainted faces. He then puts the smaller cubes back together into one larger cube such that no unpainted face is visible. Compute the number of different cubes that Dagan can make this way. Two cubes are considered the same if one can be rotated to obtain the other. You may express your answer either as an integer or as a product of prime numbers.