Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$? $\textbf{(A)}\frac{1}{7}~\textbf{(B)}\frac{1}{4}~\textbf{(C)}\frac{3}{7}~\textbf{(D)}\frac{1}{2}~\textbf{(E)}\frac{4}{7}$

In a row of $2009$ weights, the weight of each weight is an integer grams and does not exceed $1$ kg. The weights of any two adjacent weights differ by exactly $1$ g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.

Prove that for each positive integer $n$, there exists a positive integer with the following properties: [list] [*] it has exactly $n$ digits, [*] none of the digits is 0, [*] it is divisible by the sum of its digits.[/list]

Let $ABCD$ be a quadtrilateral with no parallel sides. The diagonals intersect at $E$, and $P, Q$ are points on sides $AB, CD$ respectively such that $\frac{AP}{PB} = \frac{CQ}{QD}$. $PQ$ meet $AC$ and $BD$ at $R,S$. Prove that $(EAB),(ECD),(ERS)$ all meet a point other than $E$.

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost $1$ shilling and $2$ pence, and a glass of lemonade, four sandwiches and $10$ biscuits they are worth $1$ shilling and $5$ pence. Find the price of: a) a glass of lemonade, a sandwich and a cake; b) two glasses of lemonade, three sandwiches and five biscuits. ($1$ shilling = $12$ pence).

Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$, a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ with $\alpha_n \in \{0,1\}^{p(n)}$, and a deterministic polynomial time Turing Machine $M$ such that for every $x \in \{0,1\}^n$ $$x \in L \Leftrightarrow M(x, \alpha_n) = 1$$ Let us call $\alpha_n$ to be the [i]angel string [/i]for all $x$ of the length $n$. Note that the [i]angel string[/i] is $\textbf{not}$ similar to a [i]witness[/i] or [i]certificate [/i]as used in the definition of $\textbf{NP}$ For example, all unary languages, even $UHALT$ which is undecidable, are in $\textbf{P}_{angel}$ because the \textit{angel string} can simply be a single bit that tells us if the given unary string is in $UHALT$ or not. \\\\ A set $S \subseteq \Sigma^*$ is said to be [b]sparse[/b] if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ such that for each $n \in \mathbb{N}$, the number of strings of length $n$ in $S$ is bounded by $p(n)$. In other words, $|S^{=n}| \leq p(n)$, where $S^{=n} \subseteq S$ contains all the strings in $S$ that are of length $n$. [list=1] [*] Given $k \in \mathbb{N}$ sparse sets $S_1, S_2 \ldots S_k$, show that there exists a sparse set $S$ and a deterministic polynomial time TM $M$ with oracle access to $S$ such that given an input $\langle x,i \rangle$ the TM $M$ will accept it if and only if $x \in S_i$. \\Define the set $S$ (note that it need not be computable), and give the description of $M$ with oracle $S$. \\Note that a TM $M$ with oracle access to $S$ can query whether $s \in S$ and get the correct answer in return in constant time. [/*] [*] Let us define a variant of $\textbf{P}_{angel}$ called $\textbf{P}_{bad-angel}$ with a constraint that there should exists a polynomial time algorithm that can [b]compute[/b] the angel string for any length $n \in \mathbb{N}$. In other words, there is a poly-time algorithm $A$ such that $\alpha_n = A(n)$. \\Is $\textbf{P} =\textbf{P}_{bad-angel}$? Is $\textbf{NP}=\textbf{P}_{bad-angel}$? Justify. [/*] [*] Let the language $L \in$ $\textbf{P}_{angel}$. Show that there exists a sparse set $S_L$ and a deterministic polynomial time TM $M$ with oracle access to $S_L$ that can decide the language $L$. [/*]

Let $S$ be the set of positive perfect squares that are of the form $\overline{AA}$, i.e. the concatenation of two equal integers $A$. (Integers are not allowed to start with zero.) (a) Prove that $S$ is infinite. (b) Does there exist a function $f:S\times S \rightarrow S$ such that if $a,b,c \in S$ and $a,b | c$, then $f(a,b) | c$? (If such a function $f$ exists, we call $f$ an LCM function)

$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$.

Let $S(n)$ be the sum of digits of natural number $n{}$. Is there a natural number $n{}$ for which $n+S(n)+S(S(n))=2011?$

Do either $(1)$ or $(2)$: $(1)$ A circle radius $r$ rolls around the inside of a circle radius $3r,$ so that a point on its circumference traces out a curvilinear triangle. Find the area inside this figure. $(2)$ A frictionless shell is fired from the ground with speed $v$ at an unknown angle to the vertical. It hits a plane at a height $h.$ Show that the gun must be sited within a radius $\frac{v}{g} (v^2 - 2gh)^{\frac{1}{2}}$ of the point directly below the point of impact.

For how many integers $x$ does a triangle with side lengths $10$, $24$ and $x$ have all its angles acute? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{more than } 7 $

Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.

Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.

[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends? [b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots? [b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$. [b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$. [b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions). [b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the following limit. \[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]

In a football league, there are $n\geq 6$ teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team's homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams' jerseys have at least 3 different color. Find the least number of color that the $n$ teams' $2n$ jerseys may use.

Cities $A,B,C,D$ are positioned in such a way that $A$ is closer to $C$ than to $D$, and $B$ is closer to $C$ than to $D$. Prove that every point on the straight road from $A$ to $B$ is closer to $C$ than to $D$.

Let $A$ be the number of all points with integer coordinates in a three-dimensional coordinate system. We assume that nine arbitrary points in $A$ will be colored blue. Show that we can always find two blue dots so that the line segment between them contains at least one point from $A$.

Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag? $\bullet$ Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right. $\bullet$ Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it. $\bullet$ Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

Let $ABCD$ be the rectangle. Points $E$, $F$ lies on the sides $BC$ and $CD$ respectively, such that $\sphericalangle EAF = 45^{\circ}$ and $BE = DF$. Prove that area of the triangle $AEF$ is equal to the sum of the areas of the triangles $ABE$ and $ADF$.

Six integers $a,b,c,d,e,f$ satisfy: $\begin{cases} ace+3ebd-3bcf+3adf=5 \\ bce+acf-ade+3bdf=2 \end{cases}$ Find all possible values of $abcde$ [i]D. Bazyleu[/i]