Let $\triangle ABC$ and a point $D$ on $AC$ such that $BD = DC = 3$. If $AD = 6$ and $\angle ACB = 30^\circ$, calculate $\angle ABD$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u] Set 4[/u] [b]C.16 / G.6[/b] Let $a, b$, and $c$ be real numbers. If $a^3 + b^3 + c^3 = 64$ and $a + b = 0$, what is the value of $c$? [b]C.17 / G.8[/b] Bender always turns $60$ degrees clockwise. He walks $3$ meters, turns, walks $2$ meters, turns, walks $1$ meter, turns, walks $4$ meters, turns, walks $1$ meter, and turns. How many meters does Bender have to walk to get back to his original position? [b]C.18 / G.13[/b] Guang has $4$ identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies? [b]C.19[/b] Find the sum of all digits $q$ such that there exists a perfect square that ends in $q$. [b]C.20 / G.14[/b] The numbers $5$ and $7$ are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After $2017$ minutes the product of the numbers on the board is $m$. Find the number of factors of $m$. [u]Set 5[/u] [b]C.21 / G.10[/b] On the planet Alletas, $\frac{32}{33}$ of the people with silver hair have purple eyes and $\frac{8}{11}$ of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair? [b]C.22 / G.15[/b] Let $P$ be a point on $y = -1$. Let the clockwise rotation of $P$ by $60^o$ about $(0, 0)$ be $P'$. Find the minimum possible distance between $P'$ and $(0, -1)$. [b]C.23 / G.18[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]C.24[/b] Jeremy and Kevin are arguing about how cool a sweater is on a scale of $1-5$. Jeremy says “$3$”, and Kevin says “$4$”. Jeremy angrily responds “$3.5$”, to which Kevin replies “$3.75$”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)? [b]C.25 / G.20[/b] An even positive integer $n$ has an [i]odd factorization[/i] if the largest odd divisor of $n$ is also the smallest odd divisor of $n$ greater than $1$. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u]Set 6[/u] [b]C.26 / G.26[/b] When $2018! = 2018 \times 2017 \times ... \times 1$ is multiplied out and written as an integer, find the number of $4$’s. If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \, (A/E, E/A)^3$points. [b]C.27 / G.27[/b] A circle of radius $10$ is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece. [img]https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png[/img] If the correct answer is $A$ and your answer is $E$, you will receive $\max \, \,(0, 12 - 6|A - E|)$points. [b]C.28 / G.28[/b] An equilateral triangle of side length $1$ is randomly thrown onto an infinite set of lines, spaced $1$ apart. On average, how many times will the boundary of the triangle intersect one of the lines? [img]https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png[/img] For example, in the above diagram, the boundary of the triangle intersects the lines in $2$ places. If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12-120|A-E|/A)$ points. [b]C.29 / G.29[/b] Call an ordered triple of integers $(a, b, c)$ nice if there exists an integer $x$ such that $ax^2 + bx + c = 0$. How many nice triples are there such that $-100 \le a, b, c \le 100$? If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \,(A/E, E/A)$ points. [b]C.30 / G.30[/b] Let $f(i)$ denote the number of MBMT volunteers to be born in the $i$th state to join the United States. Find the value of $1f(1) + 2f(2) + 3f(3) + ... + 50f(50)$. Note 1: Maryland was the $7$th state to join the US. Note 2: Last year’s MBMT competition had $42$ volunteers. If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12 - 500(|A -E|/A)^2)$ points. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The side lengths $a,b,c$ of a triangle $ABC$ are integers with $\gcd(a,b,c)=1$. The bisector of angle $BAC$ meets $BC$ at $D$. (a) show that if triangles $DBA$ and $ABC$ are similar then $c$ is a square. (b) If $c=n^2$ is a square $(n\ge 2)$, find a triangle $ABC$ satisfying (a).

In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds: [list] [*]Some student from $A$ knows all students from $B$. [*]Some student from $B$ knows all students from $C$. [*] Some student from $C$ knows all students from $A$.[/list]

In preparation for a game of Fish, Carl must deal $48$ cards to $6$ players. For each card that he deals, he runs through the entirety of the following process: $1$. He gives a card to a random player. $2$. A player $Z$ is randomly chosen from the set of players who have at least as many cards as every other player (i.e. $Z$ has the most cards or is tied for having the most cards). $3$. A player $D$ is randomly chosen from the set of players other than $Z$ who have at most as many cards as every other player (i.e. $D$ has the fewest cards or is tied for having the fewest cards). $4$. $Z$ gives one card to $D$. He repeats steps $1-4$ for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly $8$ cards?

Let $ABC$ be a triangle and $D$ be a point on segment $BC$. If $\triangle ABD$ is equilateral and $\angle ACB = 14^{\circ}$, what is $\angle{DAC}$? $\textbf{(A) }26^{\circ}\qquad\textbf{(B) }34^{\circ}\qquad\textbf{(C) }46^{\circ}\qquad\textbf{(D) }50^{\circ}\qquad\textbf{(E) }54^{\circ}$

Does there exist a function $f(n)$, which maps the set of natural numbers into itself and such that for each natural number $n > 1$ the following equation is satisfied $$f(n) = f(f(n - 1)) + f(f(n + 1))?$$

We have a positive integer $n$, whose sum of digits is $100$ . If the sum of digits of $44n$ is $800$ then what is the sum of digits of $3n$?

$9004$ lemmings, including an equal number of both sexes, cross in rank and file the new bridge from Eyjafjallajokull to Katla. The entire column therefore moves equidistantly and at a constant speed about the bridge, whereby this is able to hold exactly half of the lemmings for the present distance. The bridge should fulfill as many lemming dreams as possible and at the same time preserve the species be opened briefly at some point in order to halve the total population. However, the law to prevent gender discrimination requires that exact half is female. Show that these sufficient claims are also can be done. [hide=original wording]9004 Lemminge, davon gleich viele von beiden Geschlechtern, uberqueren in Reih und Glied die neue Brucke vom Eyjafjallajokull zum Katla. Die gesammte Kolonne bewegt sich also aquidistant und mit konstanter Geschwindigkeit uber die Brucke, wobei diese fur den vorliegenden Abstand genau die Halfte der Lemminge zu fassen vermag. Zur Erfullung moglichst vieler Lemmingtraume und gleichzeitiger Arterhaltung soll die Brucke irgendwann einmalig kurzzeitig aufgeklappt werden, um die Gesamtpopulation zu halbieren. Das Gesetz zur Verhinderung geschlechtsspezifischer Diskriminierung erfordert jedoch, dass davon ex akt die Halfte weiblich ist. Man zeige, dass diesen Anspruchen auch Genuge getan werden kann.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/f/8/3bf1ef0f90d3eb3761ca3db04ed48480c8aab5.png[/img]

Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$. [i]2021 CCA Math Bonanza Lightning Round #2.4[/i]

Prove for the sidelengths $a,b,c$ of a triangle $ABC$ the inequality $\frac{a^3}{b+c-a}+\frac{b^3}{c+a-b}+\frac{c^3}{a+b-c}\ge a^2+b^2+c^2$

For each even positive integer $x$, let $g(x)$ denote the greatest power of $2$ that divides $x$. For example, $g(20)=4$ and $g(16)=16$. For each positive integer $n$, let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than $1000$ such that $S_n$ is a perfect square.

Show that for each $n \ge 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a, b\in S$.

Alice and Bob play the following game: starting with the number $2$ written on a blackboard, each player in turn changes the current number $n$ to a number $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{22...2}_{2020}$. Assuming perfect play, who will win the game.

In an urban area whose street plan is a grid, a person started walking from an intersection and turned right or left at every intersection he reached until he ended up in the same initial intersection. [b]a)[/b] Show that the number of intersections (not necessarily distinct) in which he were is equivalent to $ 1 $ modulo $ 4. $ [b]b)[/b] Enunciate and prove a reciprocal statement. [i]Marius Beceanu[/i]

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? $\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4$

What is the smallest integral value of $k$ such that \[ 2x(kx-4)-x^2+6=0 \] has no real roots? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.

Let $a_0, a_1,\dots, a_{19} \in \mathbb{R}$ and $$P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}.$$ If $P(x)=P(-x)$ for all $x \in \mathbb{R}$, and $$P(k)=k^2,$$ for $k=0, 1, 2, \dots, 9$ then find $$\lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}.$$

Let $ ABC$ be an acute angled triangle whose circumcenter is $ O$. The three diameters of the circumcircle that pass through $ A$, $ B$, and $ C$, meet the opposite sides $ BC$, $ CA$, and $ AB$ at the points $ A_1$, $ B_1$ and $ C_1$, respectively. The circumradius of $ ABC$ is of length $ 2P$, where $ P$ is a prime number. The lengths of $ OA_1$, $ OB_1$, $ OC_1$ are integers. What are the lengths of the sides of the triangle?

Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

Paul fills in a $7\times7$ grid with the numbers $1$ through $49$ in a random arrangement. He then erases his work and does the same thing again, to obtain two different random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?

Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy \[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]

Which of these triples could [u]not[/u] be the lengths of the three altitudes of a triangle? $ \textbf{(A)}\ 1,\sqrt{3},2 \qquad\textbf{(B)}\ 3,4,5 \qquad\textbf{(C)}\ 5,12,13 \qquad\textbf{(D)}\ 7,8,\sqrt{113} \qquad\textbf{(E)}\ 8,15,17 $