Find the number of $16$-tuples $(x_1, x_2,..., x_{16})$ such that (i) $x_i = \pm 1$ for $i = 1,..., 16$, (ii) $0 \le x_1 + x_2 +... + x_r < 4$, for $r = 1, 2,... , 15$, (iii) $x_1 + x_2 +...+ x_{10} = 4$

Prove that between every $27$ different positive integers , less than $100$, there exist some two which are[color=red] NOT [/color]relative prime. [u]babis[/u]

How many integers less than $400$ have exactly $3$ factors that are perfect squares?

Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.

A quadrilateral $ABCD$ is inscribed in a circle and its two diagonals $AC,BD$ meet at $G$. Let $M$ be the center of $CD, E,F$ be the points on $BC, AD$ respectively such that $ME \parallel AC$ and $MF \parallel BD$. Point $H$ is the projection of $G$ onto $CD$. The circumcircle of $MEF$ meets $CD$ at $N$ distinct from $M$. Prove that $MN = MH$ Tran Quang Hung, Nguyen Le Phuoc, Thanh Xuan, Hanoi

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b\minus{}b^a\equal{}1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Let $[ABC]$ be a triangle. Points $D$, $E$, $F$ and $G$ are such $E$ and $F$ are on the lines $AC$ and $BC$, respectively, and $[ACFG]$ and $[BCED]$ are rhombus. Lines $AC$ and $BG$ meet at $H$; lines $BC$ and $AD$ meet at $I$; lines $AI$ and $BH$ meet at $J$. Prove that $[JICH]$ and $[ABJ]$ have equal area.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

The tangent lines to the circle $\omega$ at points $B$ and $D$ intersect at point $P$. The line passing through $P$ cuts out from circle chord $AC$. Through an arbitrary point on the segment $AC$ a straight line parallel to $BD$ is drawn. Prove that it divides the lengths of polygonal $ABC$ and $ADC$ in the same ratio. [hide=last sentence was in Russian: ]Докажите, что она делит длины ломаных ABC и ADC в одинаковых отношениях. [/hide]

4031 lines are drawn on a plane, no two parallel or perpendicular, and no three lines meet at a point. Determine the maximum number of acute-angled triangles that may be formed.

Let $0 \leq b \leq c \leq d \leq a$ and $a>14$ are integers. Prove, that there is such natural $n$ that can not be represented as $$n=x(ax+b)+y(ay+c)+z(az+d)$$ where $x,y,z$ are some integers. [i]K. Kohas[/i]

Let $S=\{1,2,\ldots ,98\}$. Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$-element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

Let $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ be sequences defined recursively by $a_0 =2$; $b_0 = 2$, and $a_{n+1} = a_n \sqrt{1+a_n^2+b_n^2}-b_n$; $b_{n+1} = b_n\sqrt{1+a_n^2+b_n^2} + a_n$. Find the ternary (base 3) representation of $a_4$ and $b_4$.

Suppose $n$ is a positive integer, $4n$ teams participate in a football tournament. In each round of the game, we will divide the $4n$ teams into $2n$ pairs, and each pairs play the game at the same time. After the tournament, it is known that every two teams have played at most one game. Find the smallest positive integer $a$, so that we can arrange a schedule satisfying the above conditions, and if we take one more round, there is always a pair of teams who have played in the game.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Let $x_1, x_2, \ldots, x_n$ be positive reals; for any positive integer $k$, let $S_k=x_1^k+x_2^k+\ldots+x_n^k$. (a) Given that $S_1<S_2$, show that $S_1, S_2, S_3, \ldots$ is strictly increasing. (b) Prove that there exists a positive integer $n$ and positive reals $x_1, x_2, \ldots, x_n$, such that $S_1>S_2$ and $S_1, S_2, S_3, \ldots$ is not strictly decreasing.

The satellites $A$ and $B$ circle the Earth in the equatorial plane at altitude $h$. They are separated by distance $2r$, where $r$ is the radius of the Earth. For which $h$ can they be seen in mutually perpendicular directions from some point on the equator?

Compute \[\sum_{k=1}^{420} \gcd(k,420).\] [i]2016 CCA Math Bonanza Individual #14[/i]

[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)? [b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate? [b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$ Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

Find the number of triples of real numbers $(a,b,c)$ such that $a^{20} + b^{20} + c^{20} = a^{24} + b^{24} + c^{24} =1$.