Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?

Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud. For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.

Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

We say that a tetrahedron is [i]median [/i] if and only if for each vertex the plane that passes through the midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is called [i]regular [/i] if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.

The equation \[ x^{10}+(13x-1)^{10}=0 \] has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},$ where the bar denotes complex conjugation. Find the value of \[ \frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}. \]

Consider (3-variable) polynomials \[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\] and \[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\] Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients.

For every positive real number $x$, let \[ g(x)=\lim_{r\to 0} ((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}. \] Find $\lim_{x\to \infty}\frac{g(x)}{x}$. [hide=Solution] By the Binomial Theorem one obtains\\ $\lim_{x \to \infty} \lim_{r \to 0} \left((1+r)+\frac{(1+r)r}{2}\cdot x^{-1}+\frac{(1+r)r(r-1)}{6} \cdot x^{-2}+\dots \right)^{\frac{1}{r}}$\\ $=\lim_{r \to 0}(1+r)^{\frac{1}{r}}=\boxed{e}$ [/hide]

A quadrilateral is called bicentric if it has both an incircle and a circumcircle. $ABCD$ is a bicentric quadrilateral with $(O)$ being its circumcircle. Let $E, F$ be the intersections of $AB$ and $CD, AD$ and $BC$ respectively. Prove that there is a circle with center $O$ tangent to all of the circumcircles of the four triangles $EAD, EBC, FAB, FCD$. Nguyen Van Linh, a student of the Vietnamese College, Ha Noi

Let $s(m)$ denote the sum of the digits of the positive integer $m$. Find the largest positive integer that has no digits equal to zero and satisfies the equation \[2^{s(n)} = s(n^2).\]

Show that there exists a positive constant $ c$ with the following property: To every positive irrational $ \alpha$, there can be found infinitely many fractions $ \frac{p}{q}$ with $ (p,q)\equal{}1$ satisfying $ \left|\alpha\minus{}\frac{p}{q}\right|\le \frac{c}{q^2}$

Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.

Let $k$, $\ell $ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^\circ$ and let its sidelengths be $AB=k$, $BC=\ell$ and $CD=m$. Let $f(k,\ell,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^\circ$. Prove that by fixing $\ell$ and $m$, there exists polynomial $g_{\ell,m}$ such that $f(k,\ell,m)=g_{\ell,m}(k)$ for every positive integer $k$, and find the degree of $g_{\ell,m}$ in terms of $\ell$ and $m$. [i]Submitted by Zoltán Gyenes, Budapest[/i]

Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.

Let $ABC$ be a triangle, let $O$ be its circumcenter, let $A'$ be the orthogonal projection of $A$ on the line $BC$, and let $X$ be a point on the open ray $AA'$ emanating from $A$. The internal bisectrix of the angle $BAC$ meets the circumcircle of $ABC$ again at $D$. Let $M$ be the midpoint of the segment $DX$. The line through $O$ and parallel to the line $AD$ meets the line $DX$ at $N$. Prove that the angles $BAM$ and $CAN$ are equal.

Consider the $ 4\times 4 $ integer matrices that have the property that each one of them multiplied by its transpose is $ 4I. $ [b]a)[/b] Show that the product of the elements of such a matrix is either $ 0, $ either $ 1. $ [b]b)[/b] How many such matrices have the property that the product of its elements is $ 0? $

When simplified $ \sqrt{1\plus{} \left (\frac{x^4\minus{}1}{2x^2} \right )^2}$ equals: $ \textbf{(A)}\ \frac{x^4\plus{}2x^2\minus{}1}{2x^2} \qquad \textbf{(B)}\ \frac{x^4\minus{}1}{2x^2} \qquad \textbf{(C)}\ \frac{\sqrt{x^2\plus{}1}}{2} \\ \textbf{(D)}\ \frac{x^2}{\sqrt{2}} \qquad \textbf{(E)}\ \frac{x^2}{2}\plus{}\frac{1}{2x^2}$

[b]p1.[/b] A band of pirates unloaded some number of treasure chests from their ship. The number of pirates was between $60$ and $69$ (inclusive). Each pirate handled exactly $11$ treasure chests, and each treasure chest was handled by exactly $7$ pirates. Exactly how many treasure chests were there? Show that your answer is the only solution. [b]p2.[/b] Let $a$ and $b$ be the lengths of the legs of a right triangle, let $c$ be the length of the hypotenuse, and let $h$ be the length of the altitude drawn from the vertex of the right angle to the hypotenuse. Prove that $c+h>a+b$. [b]p3.[/b] Prove that $$\frac{1}{70}< \frac{1}{2} \frac{3}{4} \frac{5}{6} ... \frac{2001}{2002} < \frac{1}{40}$$ [b]p4.[/b] Given a positive integer $a_1$ we form a sequence $a_1 , a_2 , a _3,...$ as follows: $a_2$ is obtained from $a_1$ by adding together the digits of $a_1$ raised to the $2001$-st power; $a_3$ is obtained from $a_2$ using the same rule, and so on. For example, if $a_1 =25$, then $a_2 =2^{2001}+5^{2001}$, which is a $1399$-digit number containing $106$ $0$'s, $150$ $1$'s, 4124$ 42$'s, $157$ $3$'s, $148$ $4$'s, $141$ $5$'s, $128$ $6$'s, $1504 47$'s, $152$ $8$'s, $143$ $9$'s. So $a_3 = 106 \times 0^{2001}+ 150 \times 1^{2001}+ 124 \times 2^{2001}+ 157 \times 3^{2001}+ ...+ 143 \times 9^{2001}$ which is a $1912$-digit number, and so forth. Prove that if any positive integer $a_1$ is chosen to start the sequence, then there is a positive integer $M$ (which depends on $a_1$ ) that is so large that $a_n < M$ for all $n=1,2,3,...$ [b]p5.[/b] Let $P(x)$ be a polynomial with integer coefficients. Suppose that there are integers $a$, $b$, and $c$ such that $P(a)=0$, $P(b)=1$, and $P(c)=2$. Prove that there is at most one integer $n$ such that $P(n)=4$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Decide if there are ten natural and distinct numbers $a_1,a_2,\ldots ,a_{10}$ such that: $\bullet$ Each of them is a power of a natural number with a natural exponent and greater than $1$. $\bullet$ The numbers $a_1,a_2,\ldots ,a_{10}$ form an arithmetic progression.

Grogg and Winnie are playing a game using a deck of $50$ cards numbered $1$ through $50$. They take turns with Grogg going first. On each turn a player chooses a card from the deck—this choice is made deliberately, not at random—and then adds it to one of two piles (both piles are empty at the start of the game). After all $50$ cards are in the two piles, the values of the cards in each pile are summed, and Winnie wins the positive difference of the sums of the two piles, in dollars. (For instance, if the first pile has cards summing to $510$ and the second pile has cards summing to $765$, then Winnie wins $ \$255$.) Winnie wants to win as much as possible, and Grogg wants Winnie to win as little as possible. If they both play with perfect strategy, find (with proof) the amount that Winnie wins.

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

The monic quadratic trinomial $f(x)$ has $2$ different roots. Could it be that the equation $f(f(x)) = 0$ has $3$ different root, and the equation $f(f(f(x))) = 0$ has $7$ different roots?

You own a calculator that computes exactly. It has all the standard buttons, including a button that replaces the number currently displayed with its arctangent, and a button that replaces whatever is currently displayed with its cosine. You turn on the calculator and it reads 0. You create a sequence by alternately clicking on the arctangent button and the cosine button. (The calculator is in radian mode.) Let $a_n$ be the value displayed after you've pressed the cosine button for the $n$th time. What is $\prod_{k=1}^{11} a_k$?

Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red? [i]Proposed by Matthew Weiss

Let $E$ be the set of all continuous functions $u:[0,1]\to\mathbb R$ satisfying $$u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].$$Let $\varphi:E\to\mathbb R$ be defined by $$\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.$$Prove that $\varphi$ has a maximum value and find it.

Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.