Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.

[b]p1.[/b] An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of $600$ mi/hour, and the second half of the distance is traveled at a a constant speed of $900$ mi/hour. Find the average speed of the plane. [b]p2.[/b] The figure below shows two egg cartons, $A$ and $B$. Carton $A$ has $6$ spaces (cell) and has $3$ eggs. Carton $B$ has $12$ cells and $3$ eggs. Tow cells from the total of $18$ cells are selected at random and the contents of the selected cells are interchanged. (Not that one or both of the selected cells may be empty.) [img]https://cdn.artofproblemsolving.com/attachments/6/7/2f7f9089aed4d636dab31a0885bfd7952f4a06.png[/img] (a) Find the number of selections/interchanges that produce a decrease in the number of eggs in cartoon $A$- leaving carton $A$ with $2$ eggs. (b) Assume that the total number of eggs in cartons $A$ and $B$ is $6$. How many eggs must initially be in carton $A$ and in carton $B$ so that the number of selections/interchanges that lead to an increase in the number of eggs in $A$ equals the number of selections/interchanges that lead to an increase in the number of eggs in $B$. $\bullet$ In other words, find the initial distribution of $6$ eggs between $A$ and $B$ so that the likelihood of an increase in A equals the likelihood of an increase in $B$ as the result of a selection/interchange. Prove your answer. [b]p3.[/b] Divide the following figure into four equal parts (parts should be of the same shape and of the same size, they may be rotated by different angles however they may not be disjoint and reconnected). [img]https://cdn.artofproblemsolving.com/attachments/f/b/faf0adbf6b09b5aaec04c4cfd7ab1d6397ad5d.png[/img] [b]p4.[/b] Find the exact numerical value of $\sqrt[3]{5\sqrt2 + 7}- \sqrt[3]{5\sqrt2 - 7}$ (do not use a calculator and do not use approximations). [b]p5.[/b] The medians of a triangle have length $9$, $12$ and $15$ cm respectively. Find the area of the triangle. [b]p6. [/b]The numbers $1, 2, 3, . . . , 82$ are written in an arbitrary order. Prove that it is possible to cross out $72$ numbers in such a sway the remaining number will be either in increasing order or in decreasing order. PS. You should use hide for answers.

A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

Given a triangle $ABC$ and a point $M$ such that the lines $MA,MB,MC$ intersect the lines $BC,CA,AB$ in this order in points $D,E$ and $F,$ respectively. Prove that there are numbers $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}$ such that: \[\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.\]

Two circles $\omega_1,\omega_2$ intersect at $A,B$. An arbitrary line through $B$ meets $\omega_1,\omega_2$ at $C,D$ respectively. The points $E,F$ are chosen on $\omega_1,\omega_2$ respectively so that $CE=CB,\ BD=DF$. Suppose that $BF$ meets $\omega_1$ at $P$, and $BE$ meets $\omega_2$ at $Q$. Prove that $A,P,Q$ are collinear. [i]Proposed by Iman Maghsoudi[/i]

Let there be a pentagon $ABCDE$ inscribed in a circle $O$. The tangent to $O$ at $E$ is parallel to $AD$. A point $F$ lies on $O$ and it is in the opposite side of $A$ with respect to $CD$, and satisfies $AB \cdot BC \cdot DF = AE \cdot ED \cdot CF$ and $\angle CFD = 2\angle BFE$. Prove that the tangent to $O$ at $B,E$ and line $AF$ concur at one point.

Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that \[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \] [i]Hojoo Lee, Korea[/i]

If $ N > 1$, then ${ \sqrt [3] {N \sqrt [3] {N \sqrt [3] {N}}}} =$ $ \textbf{(A)}\ N^{\frac {1}{27}}\qquad \textbf{(B)}\ N^{\frac {1}{9}}\qquad \textbf{(C)}\ N^{\frac {1}{3}}\qquad \textbf{(D)}\ N^{\frac {13}{27}}\qquad \textbf{(E)}\ N$

Let $n$ be a positive integer. A mouse sits at each corner point of an $n\times n$ board, which is divided into unit squares as shown below for the example $n=5$. [asy] unitsize(5mm); defaultpen(linewidth(.5pt)); fontsize(25pt); for(int i=0; i<=5; ++i) { for(int j=0; j<=5; ++j) { draw((0,i)--(5,i)); draw((j,0)--(j,5)); }} dot((0,0)); dot((5,0)); dot((0,5)); dot((5,5)); [/asy] The mice then move according to a sequence of [i]steps[/i], in the following manner: (a) In each step, each of the four mice travels a distance of one unit in a horizontal or vertical direction. Each unit distance is called an [i]edge[/i] of the board, and we say that each mouse [i]uses[/i] an edge of the board. (b) An edge of the board may not be used twice in the same direction. (c) At most two mice may occupy the same point on the board at any time. The mice wish to collectively organize their movements so that each edge of the board will be used twice (not necessarily be the same mouse), and each mouse will finish up at its starting point. Determine, with proof, the values of $n$ for which the mice may achieve this goal.

Prove that if for angles $A,B,C$ of a triangle holds $$\sin^2 A+\sin^2 B +\sin^2 C=2$$ iff the triangle $ABC$ is right.

Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?

Let $\omega$ be a circle with centre $O$, and let $\ell$ be a line that does not intersect $\omega$. Let $P$ be an arbitrary point on $\ell$. Let $A,B$ denote the tangent points of the tangent lines from $P$. Prove that $AB$ passes through a point being independent of choosing $P$.

Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that: $\bullet$ The area of square $KLMN$ is $1$. $\bullet$ The vertices of the triangle divide three sides of the square up into these ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale). [img]https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png[/img]

Convex quadrilateral $ABCD$ with perpendicular diagonals satisfies $\angle B = \angle C = 90^\circ$, $BC=20$, and $AD=30$. Compute the square of the area of a triangle with side lengths equal to $CD$, $DA$, and $AB$. [i]2021 CCA Math Bonanza Tiebreaker Round #2[/i]

Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?

Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with p and q relatively prime. Find $p + q + r$.