Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

Prove that if the positive reals $a, b, c$ have sum $1$ then the following inequality holds $$(ab)^{ \frac54} + (bc)^{\frac54} + (ca)^{\frac54} < \frac14 .$$

Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$

How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

Suppose that the sequence $\{a_n\}$ satisfy $a_1=1$ and $a_{2k}=a_{2k-1}+a_k, \quad a_{2k+1}=a_{2k}$ for $k=1,2, \ldots$ \\Prove that $a_{2^n}< 2^{\frac{n^2}{2}}$ for any integer $n \geq 3$.

In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$. [asy]unitsize(2.5mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); //draw(O2--O1); //dot(O1); //dot(O2); draw(Q--R); label("$Q$",Q,N); label("$P$",P,dir(80)); label("$R$",R,E); //label("12",waypoint(O1--O2,0.4),S);[/asy]

1) proove for positive $a, b, c, d$ $ \frac{a}{b+c} + \frac{b}{c+d} + \frac{c}{d+a} + \frac{d}{a+b} \ge 2$

If $ x < a < 0$ means that $ x$ and $ a$ are numbers such that $ x$ is less than $ a$ and $ a$ is less than zero, then: $ \textbf{(A)}\ x^2 < ax < 0 \qquad\textbf{(B)}\ x^2 > ax > a^2 \qquad\textbf{(C)}\ x^2 < a^2 < 0$ $ \textbf{(D)}\ x^2 > ax\text{ but }ax < 0 \qquad\textbf{(E)}\ x^2 > a^2\text{ but }a^2 < 0$

If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.

There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon.

If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.

Given a positive integer $n \geq 2$ and real numbers $a_1, a_2, \ldots, a_n \in [0,1]$. Prove that there exist real numbers $b_1, b_2, \ldots, b_n \in \{0,1\}$, such that for all $1\leq k\leq l \leq n$ we have $$\left| \sum_{i=k}^l (a_i-b_i)\right| \leq \frac{n}{n+1}.$$

Let $P(x)=x^n +a_1x^{n-1}+a_2x^{n-2}+\ldots+a_{n-1}x+a_n$ be a polynomial of degree $n$ and $n$ real roots, all of them in the interval $(0,1)$. Prove that for all $k=\overline{1,n}$ the following inequality holds: \[(-1)^k(a_k+a_{k+1}+\ldots+a_n)>0.\] [i]Proposed by N. Safaei (Iran)[/i]

$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. $(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.

The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the triangle $ABC$ respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the triangles $DEF$, $DEA$, $DBF$, $CEF$, respectively. Prove that $$d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$$ When the equality takes place?

The product of positive numbers $x, y$ and $z$ is equal to $1$. Prove that if it holds that $$\frac1x +\frac1y + \frac1z \ge x + y + z,$$ then for any natural $k$, holds the inequality $$\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.$$

Prove the inequality $\left(1+\frac{1}{q}\right)\left(1+\frac{1}{q^2}\right)...\left(1+\frac{1}{q^n}\right)<\frac{q-1}{q-2}$ for $n\in N, q>2$

Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

$x$, $y$, and $z$ are positive real numbers satisfying the equation $x+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$. Prove the following inequality: $xy + yz + zx \geq 3$.

$ABCD$ is a square with sides of unit length. Points $E$ and $F$ are taken on sides $AB$ and $AD$ respectively so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. What is this maximum area?

Let $a,b,c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge \frac12 (1+abc)$$

Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying \[ a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n} \] for all $n\geq 2024^{2024}$ is bounded. [i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

Let $D:=\{(x, y)\mid x,y\in\mathbb R_+,x \ne y,x^y=y^x\}$. (Obvious that $x\ne1$ and $y\ne1$). And let $\alpha\le\beta$ be positive real numbers. Find $$\inf_{(x,y)\in D}x^\alpha y^\beta.$$ [i]Proposed by Arkady Alt[/i]