Prove that there is no set of integers $m, n, p$ except $0, 0, 0$ for which \[m + n \sqrt2 + p \sqrt3 = 0.\]

The Rational Unit Jumping Frog starts at $(0, 0)$ on the Cartesian plane, and each minute jumps a distance of exactly $1$ unit to a point with rational coordinates. (a) Show that it is possible for the frog to reach the point $\left(\frac15,\frac{1}{17}\right)$ in a finite amount of time. (b) Show that the frog can never reach the point $\left(0,\frac14\right)$.

$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.

Let $\mathcal{H}$ be the region of points $(x, y)$, such that $(1, 0), (x, y), (-x, y)$, and $(-1,0)$ form an isosceles trapezoid whose legs are shorter than the base between $(x, y)$ and $(-x,y)$. Find the least possible positive slope that a line could have without intersecting $\mathcal{H}$.

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements. [i]***[/i]

Let $S$ be a circle, and $\alpha =\{A_1,\ldots ,A_n\}$ a family of open arcs in $S$. Let $N(\alpha )=n$ denote the number of elements in $\alpha$. We say that $\alpha$ is a covering of $S$ if $\bigcup_{k=1}^n A_k\supset S$. Let $\alpha=\{A_1,\ldots ,A_n\}$ and $\beta =\{B_1,\ldots ,B_m\}$ be two coverings of $S$. Show that we can choose from the family of all sets $A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,$ a covering $\gamma$ of $S$ such that $N(\gamma )\le N(\alpha)+N(\beta)$.

A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.

Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$. [i]Ray Li.[/i]

Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).

Let $ I $ be the center of the incircle of a triangle $ ABC. $ Shw that, if for any point $ M $ on the segment $ AB $ (extremities excluded) there exist two points $ N,P $ on $ BC, $ respectively, $ AC $ (both excluding the extremities) such that the center of mass of $ MNP $ coincides with $ I, $ then $ ABC $ is equilateral.

Find the first integer $n > 1$ such that the average of $1^2 , 2^2 ,\cdots, n^2$ is itself a perfect square.

Let square $ABCD$ and circle $\Omega$ be on the same plane, and $AA'$, $BB'$, $CC'$, $DD'$ be tangents to $\Omega$. Let $WXY Z$ be a convex quadrilateral with side lengths $WX = AA'$, $XY = BB'$, $Y Z = CC'$, and $ZW = DD'$. If $WXY Z$ has an inscribed circle, prove that the diagonals $WY$ and $XZ$ are perpendicular to each other.

In the triangle $ABC$, $\angle C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.

The three positive integers $a, b, c$ satisfy the equalities $gcd(ab, c^2) = 20$, $gcd(ac, b^2) = 18$, and $gcd(bc, a^2) = 75$. Compute the minimum possible value of $a + b + c$.

Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]

Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $\overline{DE}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$. [list=a] [*] Show that lines $PE$ and $QD$ meet on $\omega$. [*] Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well. [/list] [i]Merlijn Staps[/i]

Let $ABC$ be an acute triangle with $AB < AC$, and let $BE$ and $CF$ be the altitudes. Let the median $AM$ intersect $BE$ at point $P$, and let line $CP$ intersect $AB$ at point $D$ (see Figure 2). Prove that $DE \parallel BC$, and $AC$ is tangent to the circumcircle of $\vartriangle DEF$. [img]https://cdn.artofproblemsolving.com/attachments/f/7/bbad9f6019a77c6aa46c3a821857f06233cb93.png[/img]

Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be [i]acceptable[/i] if any two columns agree on less than $2^n-1$ entries on the same row. Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.

Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that: $$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$

The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$. (Boris Frenkin)

Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

Prove that there are infinitely many positive integers $n$ for which it is possible for a knight, starting at one of the squares of an $n \times n$ chessboard, to go through each of the squares exactly once.

$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In $2$ minutes, they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A'$s speed to $B'$s speed is: $\textbf{(A)}\ 4:5 \qquad\textbf{(B)}\ 5:6 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 5:8 \qquad\textbf{(E)}\ 1:2$

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].