Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.

Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.

Compute \[ (2+0\cdot 2 \cdot 1)+(2+0-2) \cdot (1) + (2+0)\cdot (2-1) + (2) \cdot \left(0+2^{-1}\right). \] [i]2021 CCA Math Bonanza Lightning Round #1.1[/i]

Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a+b+c=d$.

Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$.

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number. [i]Laurentiu Panaitopol[/i]

Given $m\in\mathbb{Z}^+$. Find all natural numbers $n$ that does not exceed $10^m$ satisfying the following conditions: i) $3|n.$ ii) The digits of $n$ in decimal representation are in the set $\{2,0,1,5\}$.

Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)

Let $p$ be a natural number greater than or equal to $2$ and let $(M, \cdot)$ be a finite monoid such that $a^p \ne a$, for any $a\in M \backslash \{e\}$, where $e$ is the identity element of $M$. Show that $(M, \cdot)$ is a group.

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

Define the sequence $\{x_i\}_{i \ge 0}$ by $x_0 = x_1 = x_2 = 1$ and $x_k = \frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k > 2$. Find $x_{2013}$.

Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely? [i](Proposed by Wong Jer Ren)[/i]

$99$ different points $P_1, P_2, ..., P_{99}$ are marked on circle $O$. For each $P_i$, define $n_i$ as the number of marked points you encounter starting from $P_i$ to its antipode, moving clockwise. Prove the following inequality. $$n_1+n_2+\cdots+n_{99} \leq \frac{99\cdot 98}{2}+49=4900$$

Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)

Calculate the sum of $n$ addends $$7 + 77 + 777 +...+ 7... 7.$$

Evaluate $$\sum_{k=0}^n (-1)^k \binom{n}{k} (x-k)^n.$$

For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$. [b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.

Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.

Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.