[i]a)[/i] Call a four-digit number $(xyzt)_B$ in the number system with base $B$ stable if $(xyzt)_B = (dcba)_B - (abcd)_B$, where $a \leq b \leq c \leq d$ are the digits of $(xyzt)_B$ in ascending order. Determine all stable numbers in the number system with base $B.$ [i]b)[/i] With assumptions as in [i]a[/i], determine the number of bases $B \leq 1985$ such that there exists a stable number with base $B.$

Suppose $a,b,c\in \mathbb R^+$. Prove that :\[\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\]

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

Let $S$ be the set of all convex cyclic heptagons in the plane. Define a function $f:S \rightarrow \mathbb{R}^+$, such that for any convex cyclic heptagon $ABCDEFG,$ $$f(ABCDEFG)=\frac{AC \cdot BD \cdot CE \cdot DF \cdot EG \cdot FA \cdot GB} {AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FG \cdot GA}. $$ a) Show that for any $M \in S$, $f(M) \geq f(\prod)$, where $\prod$ is a regular heptagon. b) If $f(M)=f(\prod)$, is it true that $M$ is a regular heptagon?

Let $n$ be a positive integer. Consider an infinite checkered board. A set $S$ of cells is [i]connected[/i] if one may get from any cell in $S$ to any other cell in $S$ by only traversing edge-adjacent cells in $S$. Find the largest integer $k_n$ with the following property: in any connected set with $n$ cells, one can find $k_n$ disjoint pairs of adjacent cells (that is, $k_n$ disjoint dominoes). [i]Proposed by David Anghel and Vlad Spătaru[/i]

Let $n \geq 4$ be a natural number. The polynomials $x^{n+1} + x$, $x^n$, and $x^{n-3}$ are written on the board. In one move, you can choose two polynomials $f(x)$ and $g(x)$ (not necessarily distinct) and add the polynomials $f(x)g(x)$, $f(x) + g(x)$, and $f(x) - g(x)$ to the board. Find all $n$ such that after a finite number of operations, the polynomial $x$ can be written on the board.

Consider a tetrahedron$ A_1A_2A_3A_4$. A point $N$ is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane $\alpha(N)$ (called Servais plane). Prove that if all the six points $Nij$ symmetric to a point $M$ with respect to the midpoints $Bij$ of the edges $A_iA_j$ are Servais points, then $M$ is contained in all Servais planes $\alpha(Nij )$

Given real numbers $x$, $y$, $z$ such that $xyz=-1$, show that $x^4+y^4+z^4+3(x+y+z) \ge \sum_{sym} \frac{x^2}{y}$.

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

Find the $x\in \mathbb{R}\setminus \mathbb{Q}$ such that \[ x^2+x\in \mathbb{Z}\text{ and }x^3+2x^2\in\mathbb{Z} \]

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

Solve the equation $x\cdot [x\cdot [x \cdot [x]]] = 88$ in the set of real numbers.

Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$.

Let $ f : \mathbb{N} \mapsto \mathbb{N}$ be such that [b](i)[/b] $ f$ is strictly increasing; [b](ii)[/b] $ f(mn) \equal{} f(m)f(n) \quad \forall m, n \in \mathbb{N};$ and [b](iii)[/b] if $ m \neq n$ and $ m^n \equal{} n^m,$ then $ f(m) \equal{} n$ or $ f(n) \equal{} m.$ Determine $ f(30).$

Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

Inside the regular $n$ -gon $M$ with side $a$ there are $n$ equal circles so that each touches two adjacent sides of the polygon $M$ and two other circles. Inside the formed "star", which is bounded by arcs, these $n$ equal circles are reconstructed so that each touches the two adjacent circles built in the previous step, and two more newly built circles. This process will take $k$ steps. Find the area $S_n (k)$ of the "star", which is formed in the center of the polygon $M$. Consider the spatial analogue of this problem.

We have a board of $ 2011 \times 2011$, divided by lines parallel to the edges into $1 \times 1$ squares. Manuel, Reinaldo and Jorge (at that time order) play to form squares with vertices at the vertices of the grid. The one who forms the last possible square wins, so that its sides do not cut the sides of any unit square. Who can be sure that he will win?

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

A bean packing plant has a machine that puts a certain amount of beans into bags and then puts a certain amount of bags into boxes, which are then shipped to customers. One day, the machine broke down and the first n bags came out empty, the next $n$ bags came out with $1$ bean, the next $n$ bags came out with $2$ beans,..., and the last $n$ bags came out with $2006$ beans. To provide each customer with the agreed quantity of bags of beans, the person responsible for the unit intends to distribute the bags among the $2007$ boxes that day so that all boxes contain the same number of bags and all boxes contain the same number. number of beans. For what values of $n$ is this possible?

Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.

A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 36$

Given that $x$ is a real number, find the minimum value of $f(x)=|x+1|+3|x+3|+6|x+6|+10|x+10|.$ [i]Proposed by Yannick Yao[/i]

Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .

[b]p25.[/b] Compute: $$\frac{ \sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+1)!}}{\sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+3)!}}$$ [b]p26.[/b] Suppose points $A, B, C, D$ lie on a circle $\omega$ with radius $4$ such that $ABCD$ is a quadrilateral with $AB = 6$, $AC = 8$, $AD = 7$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are respectively the angle bisectors of $\angle BAC$ and $\angle DAC$. Compute the area of quadrilateral $AECF$. [b]p27.[/b] Let $P(x) = x^2 - ax + 8$ with a a positive integer, and suppose that $P$ has two distinct real roots $r$ and $s$. Points $(r, 0)$, $(0, s)$, and $(t, t)$ for some positive integer t are selected on the coordinate plane to form a triangle with an area of $2021$. Determine the minimum possible value of $a + t$. [b]p28.[/b] A quartic $p(x)$ has a double root at $x = -\frac{21}{4}$ , and $p(x) - 1344x$ has two double roots each $\frac14$ less than an integer. What are these two double roots? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].