A dragon gave a captured knight $100$ coins. Half of them are magical, but only dragon knows which are. Each day, the knight should divide the coins into two piles (not necessarily equal in size). The day when either magic coins or usual coins are spread equally between the piles, the dragon set the knight free. Can the knight guarantee himself a freedom in at most (a) $50$ days? (b) $25$ days?

Compute the number of solutions to $1+\cos(\theta)+\cos(2\theta)+\ldots+\cos(2024\theta) = \tfrac{1}{2}$ for $\theta \in [0,2\pi].$

If $a$ is a root of $x^3-x-1 = 0$, compute the value of $$a^{10 }+ 2a^8 -a^7 - 3a^6 - 3a^5 + 4a^4 + 2a^3 - 4a^4 - 6a - 17.$$

In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.

In a country there are infinitely many towns and for every pair of towns there is one road connecting them. Initially there are $n$ coin in each city. Every day traveller Hong starts from one town and moves on to another, but if Hong goes from town $A$ to $B$ on the $k$-th day, he has to send $k$ coins from $B$ to $A$, and he can no longer use the road connecting $A$ and $B$. Prove that Hong can't travel for more than $n+2n^\frac{2}{3}$ days.

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

For positive integers $N$ and $m$, where $m \le N$, define $$a_{m,N} =\frac{1}{{N+1 \choose m}} \sum_{i=m-1}^{N-1} \frac{ {i \choose m-1}}{N - i}$$ Compute the smallest positive integer $N$ such that $$\sum^N_{m=1}a_{m,N} >\frac{2020N}{N +1}$$

Let $G$ be a graph on $n$ vertices $V_1,V_2,\dots, V_n$ and let $P_1,P_2, \dots, P_n$ be points in the plane. Suppose that, whenever $V_i$ and $V_j$ are connected by an edge, $P_iP_j$ has length $1$; in this situation, we say that the $P_i$ form an [i]embedding[/i] of $G$ in the plane. Consider a set $S\subseteq \{1,2,\dots, n\}$ and a configuration of points $Q_i$ for each $i\in S$. If the number of embeddings of $G$ such that $P_i=Q_i$ for each $i\in S$ is finite and nonzero, we say that $S$ is a [i]tasty[/i] set. Out of all tasty sets $S$, we define a function $f(G)$ to be the smallest size of a tasty set. Let $T$ be the set of all connected graphs on $n$ vertices with $n-1$ edges. Choosing $G$ uniformly and at random from $T$, let $a_n$ be the expected value of $\frac{f(G)^2}{n^2}$. Compute $\left\lfloor 2019 \displaystyle\lim_{n\to \infty} a_n \right\rfloor$. [i]Proposed by Vincent Huang[/i]

Let $ABC$ be an acute scalene triangle. Let $D, E$ be points on segments $AB, AC$ respectively, such that $BD=CE$. Prove that the nine-point centers of $ADE$, $ACD$, $ABC$, $AEB$ form a rhombus. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

Let $f: \mathbb{R}\to \mathbb{R}$ be a real function. Prove or disprove each of the following statements. (a) If f is continuous and range(f)=$\mathbb{R}$ then f is monotonic (b) If f is monotonic and range(f)=$\mathbb{R}$ then f is continuous (c) If f is monotonic and f is continuous then range(f)=$\mathbb{R}$

Show that the polynomial over variables $x,y,z$ \[ x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y) \] can't be written as a finite sum of squares of real polynomials over $x,y,z$.

The base $4$ repeating decimal $0.\overline{12}_4$ can be expressed in the form $\frac{a}{b}$ in base 10, where $a$ and $b$ are relatively prime positive integers. Compute the sum of $a$ and $b$. [i]2020 CCA Math Bonanza Team Round #2[/i]

(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible. (b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.

A set of $12$ tokens ---- $3$ red, $2$ white, $1$ blue, and $6$ black ---- is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $ \textbf{(A) }387 \qquad \textbf{(B) }388 \qquad \textbf{(C) }389 \qquad \textbf{(D) }390 \qquad \textbf{(E) }391 \qquad $

Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$

There are $n$ players in a round-robin ping-pong tournament (i.e. every two persons will play exactly one game). After some matches have been played, it is known that the total number of matches that have been played among any $n-2$ people is equal to $3^k$ (where $k$ is a fixed integer). Find the sum of all possible values of $n$.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute the sum $2019 + 201 + 20 + 2$. [b]p2.[/b] The sequence $100, 102, 104,..., 996$ and $998$ is the sequence of all three-digit even numbers. How many three digit even numbers are there? [b]p3.[/b] Find the units digit of $25\times 37\times 113\times 22$. [b]p4.[/b] Samuel has a number in his head. He adds $4$ to the number and then divides the result by $2$. After doing this, he ends up with the same number he had originally. What is his original number? [b]p5.[/b] According to Shay's Magazine, every third president is terrible (so the third, sixth, ninth president and so on were all terrible presidents). If there have been $44$ presidents, how many terrible presidents have there been in total? [b]p6.[/b] In the game Tic-Tac-Toe, a player wins by getting three of his or her pieces in the same row, column, or diagonal of a $3\times 3$ square. How many configurations of $3$ pieces are winning? Rotations and reflections are considered distinct. [b]p7.[/b] Eddie is a sad man. Eddie is cursed to break his arm $4$ times every $20$ years. How many times would he break his arm by the time he reaches age $100$? [b]p8. [/b]The figure below is made from $5$ congruent squares. If the figure has perimeter $24$, what is its area? [img]https://cdn.artofproblemsolving.com/attachments/1/9/6295b26b1b09cacf0c32bf9d3ba3ce76ddb658.png[/img] [b]p9.[/b] Sancho Panza loves eating nachos. If he eats $3$ nachos during the first minute, $4$ nachos during the second, $5$ nachos during the third, how many nachos will he have eaten in total after $15$ minutes? [b]p10.[/b] If the day after the day after the day before Wednesday was two days ago, then what day will it be tomorrow? [b]p11.[/b] Neetin the Rabbit and Poonam the Meerkat are in a race. Poonam can run at $10$ miles per hour, while Neetin can only hop at $2$ miles per hour. If Neetin starts the race $2$ miles ahead of Poonam, how many minutes will it take for Poonam to catch up with him? [b]p12.[/b] Dylan has a closet with t-shirts: $3$ gray, $4$ blue, $2$ orange, $7$ pink, and $2$ black. Dylan picks one shirt at random from his closet. What is the probability that Dylan picks a pink or a gray t-shirt? [b]p13.[/b] Serena's brain is $200\%$ the size of Eric's brain, and Eric's brain is $200\%$ the size of Carlson's. The size of Carlson's brain is what percent the size of Serena's? [b]p14.[/b] Find the sum of the coecients of $(2x + 1)^3$ when it is fully expanded. [b]p15. [/b]Antonio loves to cook. However, his pans are weird. Specifically, the pans are rectangular prisms without a top. What is the surface area of the outside of one of Antonio's pans if their volume is $210$, and their length and width are $6$ and $5$, respectively? [b]p16.[/b] A lattice point is a point on the coordinate plane with $2$ integer coordinates. For example, $(3, 4)$ is a lattice point since $3$ and $4$ are both integers, but $(1.5, 2)$ is not since $1.5$ is not an integer. How many lattice points are on the graph of the equation $x^2 + y^2 = 625$? [b]p17.[/b] Jonny has a beaker containing $60$ liters of $50\%$ saltwater ($50\%$ salt and $50\%$ water). Jonny then spills the beaker and $45$ liters pour out. If Jonny adds $45$ liters of pure water back into the beaker, what percent of the new mixture is salt? [b]p18.[/b] There are exactly 25 prime numbers in the set of positive integers between $1$ and $100$, inclusive. If two not necessarily distinct integers are randomly chosen from the set of positive integers from $1$ to $100$, inclusive, what is the probability that at least one of them is prime? [b]p19.[/b] How many consecutive zeroes are at the end of $12!$ when it is expressed in base $6$? [b]p20.[/b] Consider the following figure. How many triangles with vertices and edges from the following figure contain exactly $1$ black triangle? [img]https://cdn.artofproblemsolving.com/attachments/f/2/a1c400ff7d06b583c1906adf8848370e480895.png[/img] [b]p21.[/b] After Akshay got kicked o the school bus for rowdy behavior, he worked out a way to get home from school with his dad. School ends at $2:18$ pm, but since Akshay walks slowly he doesn't get to the front door until $2:30$. His dad doesn't like to waste time, so he leaves home everyday such that he reaches the high school at exactly $2:30$ pm, instantly picks up Akshay and turns around, then drives home. They usually get home at $3:30$ pm. However, one day Akshay left school early at exactly $2:00$ pm because he was expelled. Trying to delay telling his dad for as long as possible, Akshay starts jogging home. His dad left home at the regular time, saw Akshay on the way, picked him up and turned around instantly. They then drove home while Akshay's dad yelled at him for being a disgrace. They reached home at $3:10$ pm. How long had Akshay been walking before his dad picked him up? [b]p22.[/b] In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Then $\angle BOC = \angle BCD$, $\angle COD =\angle BAD$, $AB = 4$, $DC = 6$, and $BD = 5$. What is the length of $BO$? [b]p23.[/b] A standard six-sided die is rolled. The number that comes up first determines the number of additional times the die will be rolled (so if the first number is $3$, then the die will be rolled $3$ more times). Each time the die is rolled, its value is recorded. What is the expected value of the sum of all the rolls? [b]p24.[/b] Dora has a peculiar calculator that can only perform $2$ operations: either adding $1$ to the current number or squaring the current number. Each minute, Dora randomly chooses an operation to apply to her number. She starts with $0$. What is the expected number of minutes it takes Dora's number to become greater than or equal to $10$? [b]p25.[/b] Let $\vartriangle ABC$ be such that $AB = 2$, $BC = 1$, and $\angle ACB = 90^o$. Let points $D$ and $E$ be such that $\vartriangle ADE$ is equilateral, $D$ is on segment $\overline{BC}$, and $D$ and $E$ are not on the same side of $\overline{AC}$. Segment $\overline{BE}$ intersects the circumcircle of $\vartriangle ADE$ at a second point $F$. If $BE =\sqrt{6}$, find the length of $\overline{BF}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

The President of the Bank "Glavny Central" Gerasim Shchenkov announced that from January $2$, $2001$ until January $31$ of the same year, the dollar exchange rate would not go beyond the boundaries of the corridor of $27$ rubles $50$ kopecks. and $28$ rubles $30$ kopecks for the dollar. On January $2$, the rate will be a multiple of $5$ kopecks, and starting from January 3, each day will differ from the rate of the previous day by exactly $5$ kopecks. Mr. Shchenkov suggested that citizens try to guess what the dollar exchange rate will be during the specified period. Anyone who can give an accurate forecast for at least one day, he promised to give a cash prize. One interesting person lives in our house, a tireless arguer. For his passion for arguments and constant winnings, he was even nicknamed Zhora Sporos. Zhora claims that he can give such a forecast of the dollar exchange rate for every day from January 424 to January 4314, which he will surely guess at least once, if, of course, the banker strictly acts in accordance with the announced rules. Is Zhora right? Note: 1 ruble =100 kopecks [hide=original wording]10-I-7. Президент банка "Главный централ" Герасим Щенков объявил, что со 2-го января 2001 года и до 31-го января этого же года курс доллара не будет выходить за границы коридора 27 руб. 50 коп. и 28 руб. 30 коп. за доллар. 2-го января курс будет кратен 5 копейкам, а, начиная с 3-го января, каждый день будет отличаться от курса предыдущего дня ровно на 5 копеек. Господин Щенков предложил гражданам попробовать угадать, каким будет курс доллара в течение указанного периода. Тому, кто сумеет дать точный прогноз хотя бы на один день, он обещал выдать денежный приз. В нашем доме живет один интересный человек, неутомимый спорщик. За страсть к спорам и постоянные выигрыши его даже прозвали Жора Спорос. Жора утверждает, что может дать такой прогноз курса доллара на каждый день со 2-го по 31-е января, что обязательно хотя бы один раз угадает, если, конечно, банкир будет строго действовать в соответствии с объявленными правилами. Прав ли Жора? [/hide]

Let $n$ be a positive integer. On a board there are written all integers from $1$ to $n$. Alina does $n$ moves consecutively: for every integer $m$ $(1 \leq m \leq n)$ the move $m$ consists in changing the sign of every number divisible by $m$. At the end Alina sums the numbers. Find this sum.

Show that any integer can be expressed as a sum of two squares and a cube.

We call a real number $x$ with $0 < x < 1$ [i]interesting[/i] if $x$ is irrational and if in its decimal writing the first four decimals are equal. Determine the least positive integer $n$ with the property that every real number $t$ with $0 < t < 1$ can be written as the sum of $n$ pairwise distinct interesting numbers.

A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.