Compute $1^4+2^4+3^4+4^4+5^4+6^4$. [i]2019 CCA Math Bonanza Tiebreaker Round #1[/i]

Given a triangle $ \triangle ABC $. Denote its incenter and orthocenter by $ I, H $, respectively. If there is a point $ K $ with $$ AH+AK = BH+BK = CH+CK $$ Show that $ H, I, K $ are collinear. [i]Proposed by Evan Chen[/i]

Given are $n^2$ points in the plane, such that no three of them are collinear, where $n \geq 4$ is the positive integer of the form $3k+1$. What is the minimal number of connecting segments among the points, such that for each $n$-plet of points we can find four points, which are all connected to each other? [i]Proposed by Alexander Ivanov and Emil Kolev[/i]

Given integer $n \geq 3$. $1, 2, \ldots, n$ were written on the blackboard. In each move, one could choose two numbers $x, y$, erase them, and write down $x + y, |x-y|$ in the place of $x, y$. Find all integers $X$ such that one could turn all numbers into $X$ within a finite number of moves.

There are $2n$ people ($n \ge 2$) sitting around the round table, with each person getting to know both with his neighbors and exactly opposite him sits a person he does not know. Prove that people can rearrange in such a way that everyone knows one of their two neighbors.

Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$.

For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$

Let $$a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1$$ The value of $\sum_{k = 1}^\infty a_k$ can be expressed as a rational number $\frac{p}{q}$ in simplest form. Find $p + q$.

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

How many ways are there to arrange the letters $A,A,A,H,H$ in a row so that the sequence $HA$ appears at least once? [i]Author: Ray Li[/i]

Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.

For which natural numbers $n\ge 2$ can the numbers from $1$ to $16$ be lined up in a square scheme so that the four row sums and the four column sums are all mutually different and divisible by $n$?

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

A cube stands on one of the squares of an infinite chessboard. On each face of the cube there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks on the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. Prove that the cube cannot cover any $5\times 5$ square.

What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$? $ \textbf{(A) }2^{1002}\qquad\textbf{(B) }2^{1003}\qquad\textbf{(C) }2^{1004}\qquad\textbf{(D) }2^{1005} \qquad\textbf{(E) }2^{1006} \qquad $

Consider $ 16 $ pairwise distinct natural numbers smaller than $ 1597. $ [b]a)[/b] Prove that among these, there are three numbers having the property that the sum of any two of them is bigger than the third. [b]b)[/b] If one of these numbers is $ 1597, $ is still true the fact from subpoint [b]a)[/b]? [i]Teodor Radu[/i]

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

Let $a$ and $b$ be nonnegative real numbers. Prove that \[\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right) \le 3(a^2+b^2).\]

Many people use the social network "BWM". It is known that: By choosing any four people of that network there always is one that is a friend of the other three. Is it then true that by choosing any four people there always is one that is a friend of everyone in "BWM"? [b]Note:[/b] If member $A$ is a friend of member $B$, then member $B$ also is a friend of member $A$.

Let be two natural numbers $ n\ge 2, k, $ and $ k\quad n\times n $ symmetric real matrices $ A_1,A_2,\ldots ,A_k. $ Then, the following relations are equivalent: $ 1)\quad \left| \sum_{i=1}^k A_i^2 \right| =0 $ $ 2)\quad \left| \sum_{i=1}^k A_iB_i \right| =0,\quad\forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right) $ $ || $ [i]denotes the determinant.[/i]

Let $t$ be a real number, and let $$a_n=2\cos \left(\frac{t}{2^n}\right)-1,\quad n=1,2,3,\dots$$ Let $b_n$ be the product $a_1a_2a_3\cdots a_n$. Find a formula for $b_n$ that does not involve a product of $n$ terms, and deduce that $$\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}$$

$6$ people stand in a circle with water guns. Each person randomly selects another person to shoot. What is the probability that no pair of people shoots at each other?

$ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$.

One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set {$1,2,3$}, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly $200$; else it is successful. Prove that the sages can create a strategy, by which they can win the game.