A ball of mass $m$ is launched into the air. Ignore air resistance, but assume that there is a wind that exerts a constant force $F_0$ in the -$x$ direction. In terms of $F_0$ and the acceleration due to gravity $g$, at what angle above the positive $x$-axis must the ball be launched in order to come back to the point from which it was launched? $ \textbf{(A)}\ \tan^{-1}(F_0/mg)$ $\textbf{(B)}\ \tan^{-1}(mg/F_0)$ $\textbf{(C)}\ \sin^{-1}(F_0/mg)$ $\textbf{(D)}\ \text{the angle depends on the launch speed}$ $\textbf{(E)}\ \text{no such angle is possible}$

Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at $A$, $B$, and $C$. Compute the area of triangle $ABC$. [i]2021 CCA Math Bonanza Team Round #6[/i]

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice? [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.) [b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$ [b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square? (Same letters denote the same digits). [b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle. [img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Answer either (i) or (ii). (i) Given that the sequence whose $n$th term is $(s_n + 2s_{n + 1})$ converges, show that the sequence $\{ s_n \}$ converges also. (ii) A plane varies so that it includes a cone of constant value equal to $\pi a^3 / 3$ with the surface the equation of which in rectangular coordinates is $2xy = z^2.$ Find the equation of the envelope of the various positions of this plane. State the result so that it applies to a general cone (that is, conic surface) of the second order.

Determine all integers $n>1$ whose positive divisors add up to a power of $3.$ [i]Andrei Bâra[/i]

Consider the sequence of consecutive even numbers starting from 0, arranged in a staggered format, where each row contains one more number than the previous row. The beginning of this arrangement is shown below: $0$ $2\; 4$ $6\;\underline{8}\;10$ $12\: 14\: 16\: 18$ $20\: 22 \: 24 \: 26\: 28 $ The number in the middle of the third row is 8. What is the number in the middle of the 101-st row?

On the sides $AB$ and $BC$ of triangle $ABC$, points $K$ and $M$ are chosen such that the quadrilaterals $AKMC$ and $KBMN$ are cyclic , where $N = AM \cap CK$ . If these quads have the same circumradii, find $\angle ABC$

$\Omega$ is a quarter-circle of radius $1$. Let $O$ be the center of $\Omega$, and $A$ and $B$ be the endpoints of its arc. Circle $\omega$ is inscribed in $\Omega$. Circle $\gamma$ is externally tangent to $\omega$ and internally tangent to $\Omega$ on segment $OA$ and arc $AB$. Determine the radius of $\gamma$.

The remainder when $x^{100} -x^{99} +... -x +1$ is divided by $x^2 -1$ can be written in the form $ax +b$. Find $2a +b$. [i]Proposed by Calvin Garces[/i]

Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\frac12$ , and otherwise, if he flips tails (T), he writes $-\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\frac14$ , and otherwise he writes $-\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \frac{1}{2n}$ , and otherwise he writes $- \frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\frac17$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.

A square board with a size of $2020 \times 2020$ is divided into $2020^2$ small squares of size $1 \times 1$. Each of these small squares will be coloured black or white. Determine the number of ways to colour the board such that for every $2\times 2$ square, which consists of $4$ small squares, contains $2$ black small squares and $2$ white small squares.

Let $ABC$ be a triangle such that $\angle A=90$ and $\angle B < \angle C$. The tangent at $A$ to its circumcircle $\Gamma$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ be the midpoint of $AX$. Let the line $BY$ meet $\Gamma$ again at $Z$. Prove that the line $BD$ is tangent to circumcircle of triangle $ADZ$ .

In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.

A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to $58^{\circ}$, one of two remaining angles is equal to $59^{\circ}$, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.

Positive real numbers $a_1>a_2>\ldots>a_n$ with sum $s$ are such that the equation $nx^2-sx+1=0$ has a positive root $a_{n+1}$ smaller than $a_n$. Prove that there exists a positive integer $r \leq n$ such that the inequality $a_ra_{r+1} \geq \frac{1}{r}$ holds. [i]M. Zorka[/i]

Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that \[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.

In how many ways can we form three teams of four players each from a group of $12$ participants?

[u]Linear? What's The Problem?[/u] A function $f(x_1, x_2,..., x_n)$ is said to be linear in each of its variables if it is a polynomial such that no variable appears with power higher than one in any term. For example, $1 + x + xy$ is linear in $x$ and $y$, but $1 + x^2$ is not. Similarly, $2x + 3yz$ is linear in $x$, $y$, and $z$, but $xyz^2$ is not. [b]p8.[/b] A function $f(x,y)$ is linear in $x$ and in $y$. $f(x,y) =\frac{1}{xy}$ for $x,y \in \{3, 4\}$. What is $f(5,5)$? [b]p9.[/b] A function $f(x, y,z)$ is linear in $x$, $y$, and $z$ such that $f(x,y, z) = \frac{1}{xyz}$ for $x,y,z \in \{3,4\}$. What is $f(5, 5, 5)$? [b]p10.[/b] A function $f(x_1, x_2,..., x_n)$ is linear in each of the $x_i$ and $f(x_1, x_2,..., x_n)= \frac{1}{x_1x_2...x_n}$ when $x_i \in \{3,4\}$ for all $ i$. In terms of $n$, what is $f(5,5,...,5)$?

Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system: $$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$ Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $ [i]Florian Dumitrel[/i]

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$. Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.

Show that there are no positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ such that $$(1+a_1 \omega)(1+a_2 \omega)(1+a_3 \omega)(1+a_4 \omega)(1+a_5 \omega)(1+a_6 \omega)$$ is an integer where $\omega$ is an imaginary $5$th root of unity.

Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}$, $|AB|=2$, $|AD|=4$ and $|BC|=|BD|$. If the line through $C$ which is parallel to $AB$ meets $AD$ at $E$, what is $|CE|$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None of the preceding} $

Some cities of Russia are connected with some cities of Ukraine with international airlines. The Interstate Council for the Promotion of Migration intends to introduce a one-way traffic on each airline so that, by taking off from the city, it could no longer be returned in this city (using other one-way airlines). Prove that the number of ways to do this is not divided by $3$.

Alphonse and Beryl play a game involving $n$ safes. Each safe can be opened by a unique key and each key opens a unique safe. Beryl randomly shuffles the $n$ keys, and after placing one key inside each safe, she locks all of the safes with her master key. Alphonse then selects $m$ of the safes (where $m < n$), and Beryl uses her master key to open just the safes that Alphonse selected. Alphonse collects all of the keys inside these $m$ safes and tries to use these keys to open up the other $n - m$ safes. If he can open a safe with one of the $m$ keys, he can then use the key in that safe to try to open any of the remaining safes, repeating the process until Alphonse successfully opens all of the safes, or cannot open any more. Let $P_m(n)$ be the probability that Alphonse can eventually open all $n$ safes starting from his initial selection of $m$ keys. (a) Show that $P_2(3) = \frac23$. (b) Prove that $P_1(n) = \frac1n$. (c) For all integers $n \geq 2$, prove that $$P_2(n) = \frac2n \cdot P_1(n-1) + \frac{n-2}{n} \cdot P_2(n-1).$$ (d) Determine a formula for $P_2 (n)$.