Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? $ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$

A two-digit positive integer is said to be [i]cuddly[/i] if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Suppose $a$, $b$, and $c$ are nonzero real numbers such that \[bc+\frac1a = ca+\frac2b = ab+\frac7c = \frac1{a+b+c}.\] Find $a+b+c$.

[b]p1.[/b] Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular. Decide whether this statement is a true or false proposition in euclidean geometry. If it is true, prove it; if false, produce a counterexample. [b]p2.[/b] Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value between $1$ and $5$, for any real value of $x$. [b]p3.[/b] A man walked a total of $5$ hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks $4$ miles per hour on the level, three miles per hour uphill, and $r$ miles per hour downhill. For what values of $r$ will this information uniquely determine his total walking distance? [b]p4.[/b] A point $P$ is so located in the interior of a rectangle that the distance of $P$ from one comer is $5$ yards, from the opposite comer is $14$ yards, and from a third comer is $10$ yards. What is the distance from $P$ to the fourth comer? [b]p5.[/b] Each small square in the $5$ by $5$ checkerboard shown has in it an integer according to the following rules: $\begin{tabular}{|l|l|l|l|l|} \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline \end{tabular}$ i. Each row consists of the integers $1, 2, 3, 4, 5$ in some order. ii. Тhе order of the integers down the first column has the same as the order of the integers, from left to right, across the first row and similarly fог any other column and the corresponding row. Prove that the diagonal squares running from the upper left to the lower right contain the numbers $1, 2, 3, 4, 5$ in some order. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Each of the integers from $1$ to $n$ is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, $A,B,$ and $C,$ take turns in the order $A,B,C,A,\dots$ choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered $1.$ Show that for each of the three players, there are arbitrarily large values of $n$ for which that player has the highest probability among the three players of winning the game.

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2-2xyz=1$. Prove that \[ (1+x)(1+y)(1+z)\le 4+4xyz. \]

Evaluate $$ \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1\slash x},$$ where $a>0$ and $a\ne 1.$

If $a_i >0$ ($i=1, 2, \cdots , n$), then $$\left (\frac{a_1}{a_2} \right )^k + \left (\frac{a_2}{a_3} \right )^k + \cdots + \left (\frac{a_n}{a_1} \right )^k \geq \frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots + \frac{a_n}{a_1}$$ for all $k\in \mathbb{N}$

Divide a regular $8960$-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$?

2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.

Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$

Let $f:[0,1]\to(0,1)$ be a surjective function. [list=a] [*]Prove that $f$ has at least one point of discontinuity. [*]Given that $f$ admits a limit in any point of the interval $[0,1],$ show that is has at least two points of discontinuity. [/list][i]Mihai Piticari and Sorin Rădulescu[/i]

For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$. Define the sequence $a_1, a_2, a_3,...$ as followed: $a_1> 1$ is an arbitrary positive integer, $a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$. Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.

A ladder $10\text{ m}$ long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $1\text{ m/s}$, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is $6\text{ m}$ from the wall?

Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?

A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? $\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$

Given a positive integer $n$. A triangular array $(a_{i,j})$ of zeros and ones, where $i$ and $j$ run through the positive integers such that $i+j\leqslant n+1$ is called a [i]binary anti-Pascal $n$-triangle[/i] if $a_{i,j}+a_{i,j+1}+a_{i+1,j}\equiv 1\pmod{2}$ for all possible values $i$ and $j$ may take on. Determine the minimum number of ones a binary anti-Pascal $n$-triangle may contain.

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

Consider the triangle $ ABC$ with $ m(\angle BAC \equal{} 90^\circ)$ and $ AC \equal{} 2AB$. Let $ P$ and $ Q$ be the midpoints of $ AB$ and $ AC$,respectively. Let $ M$ and $ N$ be two points found on the side $ BC$ such that $ CM \equal{} BN \equal{} x$. It is also known that $ 2S[MNPQ] \equal{} S[ABC]$. Determine $ x$ in function of $ AB$.

The given sequences are $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for every real $ t $ the sequence $$ (x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })$$ has the following property: there is a number $ k $ such that $ 1 \leq k \leq n $ and all non-zero terms of the sequence with indices less than $ k $ are of the same sign and all non-zero terms of the sequence with indices not less than $ k $ are the same sign.

A regular hexagon $ABCDEF$ has perimeter $12$. $AB$, $CD$, and $EF$ are all extended, and the intersections of the line segments form an equilateral triangle. Compute the perimeter of the triangle.

We are given five watches which can be winded forward. What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?

Let us consider an infinite grid plane as shown below. We start with 4 points $A$, $B$, $C$, $D$, that form a square. We perform the following operation: We pick two points $X$ and $Y$ from the currant points. $X$ is reflected about $Y$ to get $X'$. We remove $X$ and add $X'$ to get a new set of 4 points and treat it as our currant points. For example in the figure suppose we choose $A$ and $B$ (we can choose any other pair too). Then reflect $A$ about $B$ to get $A'$. We remove $A$ and add $A'$. Thus $A'$, $B$, $C$, $D$ is our new 4 points. We may again choose $D$ and $A'$ from the currant points. Reflect $D$ about $A'$ to obtain $D'$ and hence $A'$, $B$, $C$, $D'$ are now new set of points. Then similar operation is performed on this new 4 points and so on. Starting with $A$, $B$, $C$, $D$ can you get a bigger square by some sequence of such operations?

There is an $n \times n$ table with $n -1$ cells containing ones and the remaining cells containing zeros. You can do this with the table the following operation: select the tap hole, subtract from the number in this cell, one, and to all other numbers on the same line or in the same column as the selected cell, add one. Is it possible from of this table, using the specified operations, obtain a table in which all numbers are equal?

In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]