Let $x,y,z$ be real numbers. Find the minimum value of the sum \begin{align*}|\cos(x)|+|\cos(y)|+|\cos(z)|+|\cos(x-y)|+|\cos(y-z)|+|\cos(z-x)|.\end{align*}

Let $C$ be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners at $C,$ keeping the new faces perpendicular to the diagonals of the cube, and keeping the newly formed faces indentical. If at the conclusion of this process the fourteen faces so have the same area, find the area of each of face of the new solid.

Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.

If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.

We are given $2021$ points on a plane, no three of which are collinear. Among any $5$ of these points, at least $4$ lie on the same circle. Is it necessarily true that at least $2020$ of the points lie on the same circle?

An ant walks a distance $A = 10^9$ millimeters. A bear walks $B = 10^6$ feet. A chicken walks $C = 10^8$ inches. What is the correct ordering of $A, B, C?$ (Note there are $25.4$ millimeters in an inch, and there are $12$ inches in a foot.) $$ \mathrm a. ~ A<B<C\qquad \mathrm b.~A<C<B\qquad \mathrm c. ~C<B<A \qquad \mathrm d. ~B<A<C \qquad \mathrm e. ~B<C<A $$

Convex polyhedron $M$ with triangular faces is cut into tetrahedrons; all the vertices of the tetrahedrons are the vertices of the polyhedron, and any two tetrahedrons either do not intersect, or they intersect along a common vertex, common edge, or common face. Prove that it it's not possible that each tetrahedron has exactly one face on the surface of $M$.

Let $ABCD$ be a tetrahedron and let $d$ be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most $\frac{\sqrt{d}}{2\sqrt{3}}$

This year February $29$ fell on a Sunday. In what year will February $29$ next fall on a Sunday?

The lines \(AB\) and \(CD\), containing the lateral sides of the trapezoid \(ABCD\), intersect at point \(Q\). Inside the trapezoid \(ABCD\), a point \(P\) is chosen such that \(\angle APB = \angle CPD\). Prove that the circumcircles of triangles \(BPD\) and \(APC\) intersect again on the line \(PQ\). [i]Proposed by Mykhailo Shtandenko[/i]

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that \[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]

A set of $n$ lines are said to be in [i]standard form[/i] if no two are parallel and no three are concurrent. Does there exist a value of $k$ such that given any $n$ lines in [i]standard form[/i], it is possible to colour the regions bounded by the $n$ lines using $k$ colours in such a way that no two regions of the same colour share a common intersection point of the $n$ lines?

For positive integer $k,$ we say that it is a [i]Taurus integer[/i] if we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square. $(1)$ Determine whether $2021$ is a Taurus integer. $(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$

In the addition problem \[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a different digit. Find the number represented by the answer PICNIC.

Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$. [i]Author:Anderson Wang[/i]

Given $2015$ subsets $A_1, A_2,...,A_{2015}$ of the set $\{1, 2,..., 1000\}$ such that $|A_i| \ge 2$ for every $i \ge 1$ and $|A_i \cap A_j| \ge 1$ for every $1 \le i < j \le 2015$. Prove that $k = 3$ is the smallest number of colors such that we can always color the elements of the set $\{1, 2,..., 1000\}$ by $k$ colors with the property that the subset $A_i$ has at least two elements of different colors for every $i \ge 1$. Lê Anh Vinh

Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$. (a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$. (b) Determine the length of the perimeter of this hexagon when $a + b > c$.

The square $ABCD$ is inscribed in a circle with center $O$. Let $E$ be the midpoint of $AD$. The line $CE$ meets the circle again at $F$. The lines $FB$ and $AD$ meet at $H$. Prove $HD = 2AH$

In the country of PUMaC-land, there are $5$ villages and $3$ cities. Vedant is building roads between the $8$ settlements according to the following rules: a) There is at most one road between any two settlements; b) Any city has exactly three roads connected to it; c) Any village has exactly one road connected to it; d) Any two settlements are connected by a path of roads. In how many ways can Vedant build the roads?

A sequence starts with $2017$ as its first term and each subsequent term is the sum of cubes of the digits in the previous number. What is the $2017$th term of this sequence? [i]2017 CCA Math Bonanza Individual Round #3[/i]

$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.

34. Find the sum of the ages of everyone who wrote a problem for this year’s HMMT November contest. If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - |X - Y|)$ 35. Find the total number of occurrences of the digits $0, 1 \ldots , 9$ in the entire guts round (the official copy). If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - \frac{|X-Y|}{2})$ 36. Find the number of positive integers less than $1000000$ which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\text{max}(0, 20 - 80|1 - \frac{X}{Y}|)$ rounded to the nearest integer.