Find $n$ such that $20^{2009}=10^{2000}\cdot 40^9\cdot 2^n$.

Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?

[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train travels at a constant speed, what is the distance from $P$ to $Q$? [b]p2.[/b] If $a$ and $b$ are integers, $b$ odd, prove that $x^2 + 2ax + 2b = 0$ has no rational roots. [b]p3.[/b] A diameter segment of a set of points in a plane is a segment joining two points of the set which is at least as long as any other segment joining two points of the set. Prove that any two diameter segments of a set of points in the plane must have a point in common. [b]p4.[/b] Find all positive integers $n$ for which $\frac{n(n^2 + n + 1) (n^2 + 2n + 2)}{2n + 1}$ is an integer. Prove that the set you exhibit is complete. [b]p5.[/b] $A, B, C, D$ are four points on a semicircle with diameter $AB = 1$. If the distances $\overline{AC}$, $\overline{BC}$, $\overline{AD}$, $\overline{BD}$ are all rational numbers, prove that $\overline{CD}$ is also rational. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Circles $k$ and $l$ intersect at points $P$ and $Q$. Let $A$ be an arbitrary point on $k$ distinct from $P$ and $Q$. Lines $AP$ and $AQ$ meet $l$ again at $B$ and $C$. Prove that the altitude from $A$ in triangle $ABC$ passes through a point that does not depend on $A$.

Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.

Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying: (1) $|A_i|\leq 3,i=1,2,...,k$ (2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$. Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$

Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$ and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.

We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not.

Let $BE$ and $CF$ be the altitudes of an acute triangle $ABC$ with $E$ on $AC$ and $F$ on $AB$. Let $O$ be the point of intersection of $BE$ and $CF$. Take any line $KL$ through $O$ with $K$ on $AB$ and $L$ on $AC$. Suppose $M$ and $N$ are located on $BE$ and $CF$ respectively. such that $KM$ is perpendicular to $BE$ and $LN$ is perpendicular to $CF$. Prove that $FM$ is parallel to $EN$.

Four friends, Anna, Bob, Celia, and David, exchanged some money. For any two of these friends, exactly one gave money to the other. For example, Celia could have given some money to David but then David would not have given money to Celia. In the end, each person broke even (meaning that no one made or lost any money). (a) Is it possible that the amounts of money given were $10$, $20$, $30$, $40$, $50$, $60$? (b) Is it possible that the amounts of money given were $20$, $30$, $40$, $50$, $60$, $70$? For each part, if your answer is yes, show that the situation is possible by describing who could have given what amounts to whom. If your answer is no, prove that the situation is not possible.

Each of $n$ boys and $n$ girls chooses a random number from the set $\{ 1, 2, 3, 4, 5 \}$, uniformly and independently. Let $p_n$ be the probability that every boy chooses a different number than every girl. As $n$ approaches infinity, what value does $\sqrt[n]{p_n}$ approach?

Find all functions $f: \mathbb R^+ \to \mathbb R^+$ such that \[ f \left( x+\frac{f(xy)}{x} \right) = f(xy) f \left( y + \frac 1y \right) \] holds for all $x,y\in\mathbb R^+.$

Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

Real numbers $x$, $y$, and $z$ satisfy the inequalities $$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$ Which of the following numbers is nessecarily positive? $\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\ \textbf{(E) } y+z$

\[\int_0^{2022} \{x\lfloor x \rfloor\}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

Let $D,E,F$ be the points of tangency of the incircle of a given triangle $ABC$ with sides $BC, CA, AB,$ respectively. Denote by $I$ the incenter of $ABC$, by $M$ the midpoint of $BC$ and by $G$ the foot of the perpendicular from $M$ to line $EF$. Prove that the line $ID$ is tangent to the circumcircle of the triangle $MGI$.

Prove that if $n$ is an odd natural number, then $1^n +2^n +\cdots +n^n$ is divisible by $n^2$.

The roots of the equation $ax^2 - 4bx + 4c = 0$ with $ a > 0$ belong to interval $[2, 3]$. Prove that: a) $a \le b \le c < a + b.$ b) $\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .$

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]

Let $\tfrac{a}{b}$ be a fraction such that $a$ and $b$ are positive integers and the first three digits of its decimal expansion are $527$. What is the smallest possible value of $a+b?$

Given is an acute triangle $ABC$. Point $P$ lies inside this triangle and lies on the bisector of angle $\angle BAC$. Suppose that the point of intersection of the altitudes $H$ of triangle $ABP$ lies inside triangle $ABC$. Let $Q$ be the intersection of the line $AP$ and the line perpendicular to $AC$ passing through $H$. Prove that $Q$ is the point symmetrical to $P$ wrt the line $BH$.