Let be two real numbers $ a>0,b. $ Calculate the primitive of the function $ 0<x\mapsto\frac{bx-1}{e^{bx}+ax} . $ [i]Dan Negulescu[/i]

Write an integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors. Note: Two triangles are said to be neighbors if they have a common side. [asy]size(100); pair P=(0,0); pair Q=(2, 2*sqrt(3)); pair R=(4,0); draw(P--Q--R--cycle); pair B=midpoint(P--Q); pair A=midpoint(P--B); pair C=midpoint(B--Q); pair E=midpoint(Q--R); pair D=midpoint(Q--E); pair F=midpoint(E--R); pair H=midpoint(R--P); pair G=midpoint(R--H); pair I=midpoint(H--P); draw(A--I); draw(B--H); draw(C--G); draw(I--D); draw(H--E); draw(G--F); draw(C--D); draw(B--E); draw(A--F);[/asy]

In the regular hexagon shown below, how many diagonals are longer than the red diagonal? [asy] size(2cm); draw((0,0)--(2,0)--(3,1.732)--(2,3.464)--(0,3.464)--(-1,1.732)--cycle); draw((-1,1.732)--(2,0),red); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$ (a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.

Find solution in positive integers : $$n^5+n^4=7^m-1$$

A bag has $3$ white and $7$ black marbles. Arjun picks out one marble without replacement and then a second. What is the probability that Arjun chooses exactly $1$ white and $1$ black marble?

Pedro and Juan are playing the following game: $-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$). $-$ Each player can draw: -- $1$ rock from one of the piles, or -- $2$ rocks from one of the piles, or -- $1$ rock from each pile, or -- $2$ rock from one pile and $1$ from the other pile. Each player must perform one of these four operations in their turns. The looser is the one who takes the last rock. Pedro plays first and has a winning strategy. What are the three maximum possible values of ($X+Y$)?

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

Find the smallest possible $n$ for which there exist integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$ such that each integer between $1000$ and $2000$ (inclusive) can be written as the sum (without repetition), of one or more of the integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$.

Jason has n coins, among which at most one of them is counterfeit. The counterfeit coin (if there is any) is either heavier or lighter than a real coin. Jason’s grandfather also left him an old weighing balance, on which he can place any number of coins on either side and the balance will show which side is heavier. However, the old weighing balance is in fact really really old and can only be used 4 more times. What is the largest number $n$ for which it is possible for Jason to find the counterfeit coin (if it exist)?

Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$. [i]Author: Christopher Bradley, United Kingdom [/i]

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

17) An object is thrown directly downward from the top of a 180-meter-tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof? A) 15 m/s B) 25 m/s C) 35 m/s D) 55 m/s E) insufficient information

Find the sum of the coefficients of the polynomial $P(x) = x^4- 29x^3 + ax^2 + bx + c$, given that $P(5) = 11$, $P(11) = 17$, and $P(17) = 23$.

A castaway is building a rectangular raft $ABCD$. He fixes a mast perpendicular to the raft, with ropes passing from the top of the mast (point $S$ in the figure) to the four corners of the raft. The rope $SA$ measures $8$ meters, the rope $SB$ measures $2$ meters and the rope $SC$ measures $14$ meters. Compute the length of the rope $SD$. [asy] size(250); // Coordinates for the parallelogram ABCD pair A = (0, 0); pair B = (8, 0); pair C = (10, 5); pair D = (2, 5); // Position of point S (outside the parallelogram) pair S = (5, 8); pair T = (5, 3); // Draw the parallelogram ABCD filldraw(A--B--C--D--cycle, lightgray, black); // Draw the ropes from point S to each corner of the parallelogram draw(S--A, blue); draw(S--B, blue); draw(S--C, blue); draw(S--D, blue); draw(S--T, black); // Mark the points dot(A); dot(B); dot(C); dot(D); dot(S); dot(T); // Label the points label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, NW); label("S", S, N); [/asy]

(1) Show the following inequality for every natural number $ k$. \[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\] (2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$. \[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]

Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$.

N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$. I.Gorodnin

Let $k$ be a positive integer. An arrangement of finitely many open intervals in $R$ is called [i]good [/i] if for any of the intervals the number of other intervals which intersect with it is a nonzero multiple of $k$. Find the maximum positive integer $n$ (as a function of $k$) for which there is no good arrangement with $n$ intervals

For a positive integer $m$, let $\varphi(m)$ be the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime, and let $\sigma(m)$ be the sum of the positive divisors of $m$. Find the sum of all even positive integers $n$ such that \[ \frac{n^5\sigma(n) - 2}{\varphi(n)} \] is an integer.

Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.

Consider a cylinder and a cone with a common base such that the volume of the part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.

Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.