[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.

A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of $\tfrac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle. [asy] size(150); pair A = (-5,0); pair B = (5,0); pair C = (-3,4); pair D = (3,4); pair E = intersectionpoint(B--C,A--D); draw(A--B--D--cycle); draw(A--C); draw(C--E); draw(E--B,dashed); markscalefactor=0.06; draw(rightanglemark(A,C,B)); [/asy]

Find all natural numbers $n$ such that $5^n+3$ is a power of $2$

Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$ [i]Marian Andronache[/i]

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Determine all sequences of equal ratios of the form \[ \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8} \] which simultaneously satisfy the following conditions: $\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$. $\bullet$ The common value of the ratios is a natural number.

Let $[ABC]$ be a acute-angled triangle and its circumscribed circle $\Gamma$. Let $D$ be the point on the line $AB$ such that $A$ is the midpoint of the segment $[DB]$ and $P$ is the point of intersection of $CD$ with $\Gamma$. Points $W$ and $L$ lie on the smaller arcs $\overarc{BC}$ and $\overarc{AB}$, respectively, and are such that $\overarc{BW} = \overarc{LA }= \overarc{AP}$. The $LC$ and $AW$ lines intersect at $Q$. Shows that $LQ = BQ$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

There are $456$ people around a circle, denoted as $X_1, X_2, \dots, X_{456}$, and each one of them thought of a number. Every time Laura says an integer $k$ with $2 \leqslant k \leqslant 100$, the announcer announces all the numbers $p_1, p_2, \dots, p_{456}$, which are the averages of the numbers thought by the people in all the groups of $k$ consecutive people: $p_1$ is the average of the numbers thought by the people from $X_1$ to $X_k$, $p_2$ is the average of the numbers thought by the people from $X_2$ to $X_{k+1}$, and so on until $p_{456}$, the average of the numbers thought by the people from $X_{456}$ to $X_{k-1}$. Determine how many numbers $k$ Laura must say at a minimum so that, with certainty, the announcer can know the number thought by the person $X_{456}$.

The geometric mean (G.M.) of $k$ positive integers $a_1$, $a_2$, $\dots$, $a_k$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, $CA=15$. Let $O$ be the circumcenter of $ABC$. Find the distance between the circumcenters of triangles $AOB$ and $AOC$.

Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$. \[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]

Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that \[ \frac {PA}{BC^{2}} \plus{} \frac {PB}{CA^{2}} \plus{} \frac {PC}{AB^{2}}\ge \frac {1}{R}. \] [i]Alternative formulation:[/i] If $ ABC$ is a triangle with sidelengths $ BC\equal{}a$, $ CA\equal{}b$, $ AB\equal{}c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that $ \frac {PA}{a^{2}} \plus{} \frac {PB}{b^{2}} \plus{} \frac {PC}{c^{2}}\ge \frac {1}{R}$.

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

Calculate the following indefinite integrals. [1] $\int \frac{dx}{1+\cos x}$ [2] $\int x\sqrt{x^2-1}dx$ [3] $\int a^{-\frac{x}{2}}dx\ \ (a>0,a\neq 1)$ [4] $\int \frac{\sin ^ 3 x}{1+\cos x}dx$ [5] $\int e^{4x}\sin 2x dx$

Is it possible to partition the set $S=\{1,2,\ldots,21\}$ into subsets that in each of these subsets the largest number is equal to the sum of the other numbers?

Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?

There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le 240$ such that ak is a multiple of $3$.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let $ \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$ where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$.

If $2x-3y-z=0$ and $x+3y-14z=0$, $z \neq 0$, the numerical value of $\frac{x^2+3xy}{y^2+z^2}$ is: $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ -20/17\qquad\textbf{(E)}\ -2 $

Consider an angle $ \angle DEF$, and the fixed points $ B$ and $ C$ on the semiline $ (EF$ and the variable point $ A$ on $ (ED$. Determine the position of $ A$ on $ (ED$ such that the sum $ AB\plus{}AC$ is minimum.

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$