Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra. [asy] import graph; size(12.57cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.79, xmax = 8.79, ymin = 0.32, ymax = 4.18; /* image dimensions */ pen ffqqtt = rgb(1,0,0.2); pen ffzzzz = rgb(1,0.6,0.6); pen zzzzff = rgb(0.6,0.6,1); draw((6,3.5)--(8,1.5), zzzzff); draw((7,3)--(5,1), blue); draw((6,3.5)--(7,3), blue); draw((6,3.5)--(5,1), blue); draw((5,1)--(8,1.5), blue); draw((7,3)--(8,1.5), blue); draw((4,3.5)--(2,1.5), ffzzzz); draw((1,3)--(2,1.5), ffqqtt); draw((2,1.5)--(3,1), ffqqtt); draw((1,3)--(3,1), ffqqtt); draw((4,3.5)--(1,3), ffqqtt); draw((4,3.5)--(3,1), ffqqtt); draw((-3,3)--(-3,1), linewidth(1.6)); draw((-3,3)--(-1,3), linewidth(1.6)); draw((-1,3)--(-1,1), linewidth(1.6)); draw((-3,1)--(-1,1), linewidth(1.6)); draw((-3,3)--(-2,3.5), linewidth(1.6)); draw((-2,3.5)--(0,3.5), linewidth(1.6)); draw((0,3.5)--(-1,3), linewidth(1.6)); draw((0,3.5)--(0,1.5), linewidth(1.6)); draw((0,1.5)--(-1,1), linewidth(1.6)); draw((-3,1)--(-2,1.5)); draw((-2,1.5)--(0,1.5)); draw((-2,3.5)--(-2,1.5)); draw((1,3)--(1,1), linewidth(1.6)); draw((1,3)--(3,3), linewidth(1.6)); draw((3,3)--(3,1), linewidth(1.6)); draw((1,1)--(3,1), linewidth(1.6)); draw((1,3)--(2,3.5), linewidth(1.6)); draw((2,3.5)--(4,3.5), linewidth(1.6)); draw((4,3.5)--(3,3), linewidth(1.6)); draw((4,3.5)--(4,1.5), linewidth(1.6)); draw((4,1.5)--(3,1), linewidth(1.6)); draw((1,1)--(2,1.5)); draw((2,3.5)--(2,1.5)); draw((2,1.5)--(4,1.5)); draw((5,3)--(5,1), linewidth(1.6)); draw((5,3)--(6,3.5), linewidth(1.6)); draw((5,3)--(7,3), linewidth(1.6)); draw((7,3)--(7,1), linewidth(1.6)); draw((5,1)--(7,1), linewidth(1.6)); draw((6,3.5)--(8,3.5), linewidth(1.6)); draw((7,3)--(8,3.5), linewidth(1.6)); draw((7,1)--(8,1.5)); draw((5,1)--(6,1.5)); draw((6,3.5)--(6,1.5)); draw((6,1.5)--(8,1.5)); draw((8,3.5)--(8,1.5), linewidth(1.6)); label("$ A $",(-3.4,3.41),SE*labelscalefactor); label("$ D $",(-2.16,4.05),SE*labelscalefactor); label("$ H $",(-2.39,1.9),SE*labelscalefactor); label("$ E $",(-3.4,1.13),SE*labelscalefactor); label("$ F $",(-1.08,0.93),SE*labelscalefactor); label("$ G $",(0.12,1.76),SE*labelscalefactor); label("$ B $",(-0.88,3.05),SE*labelscalefactor); label("$ C $",(0.17,3.85),SE*labelscalefactor); label("$ A $",(0.73,3.5),SE*labelscalefactor); label("$ B $",(3.07,3.08),SE*labelscalefactor); label("$ C $",(4.12,3.93),SE*labelscalefactor); label("$ D $",(1.69,4.07),SE*labelscalefactor); label("$ E $",(0.60,1.15),SE*labelscalefactor); label("$ F $",(2.96,0.95),SE*labelscalefactor); label("$ G $",(4.12,1.67),SE*labelscalefactor); label("$ H $",(1.55,1.82),SE*labelscalefactor); label("$ A $",(4.71,3.47),SE*labelscalefactor); label("$ B $",(7.14,3.10),SE*labelscalefactor); label("$ C $",(8.14,3.82),SE*labelscalefactor); label("$ D $",(5.78,4.08),SE*labelscalefactor); label("$ E $",(4.6,1.13),SE*labelscalefactor); label("$ F $",(6.93,0.96),SE*labelscalefactor); label("$ G $",(8.07,1.64),SE*labelscalefactor); label("$ H $",(5.65,1.90),SE*labelscalefactor); dot((-3,3),dotstyle); dot((-3,1),dotstyle); dot((-1,3),dotstyle); dot((-1,1),dotstyle); dot((-2,3.5),dotstyle); dot((0,3.5),dotstyle); dot((0,1.5),dotstyle); dot((-2,1.5),dotstyle); dot((1,3),dotstyle); dot((1,1),dotstyle); dot((3,3),dotstyle); dot((3,1),dotstyle); dot((2,3.5),dotstyle); dot((4,3.5),dotstyle); dot((4,1.5),dotstyle); dot((2,1.5),dotstyle); dot((5,3),dotstyle); dot((5,1),dotstyle); dot((6,3.5),dotstyle); dot((7,3),dotstyle); dot((7,1),dotstyle); dot((8,3.5),dotstyle); dot((8,1.5),dotstyle); dot((6,1.5),dotstyle); [/asy]

Prove that every polygon that has a center of symmetry can be dissected into a square such that it is divided into finitely many polygonal pieces, and all the pieces can only be translated. (In other words, the original polygon can be divided into polygons $A_1,A_2,\dotsc ,A_n$, a square can be divided into polygons a $B_1,B_2,\dotsc ,B_n$ such that for $1\leqslant i\leqslant n$ polygon $B_i$ is a translated copy of polygon $A_i$.)

Screws are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$?

Prove that for positive reals $a$, $b$, $c$ we have \[ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. \]

Let $a,b,c$ be real numbers such that $$a^2-bc=b^2-ca=c^2-ab=2$$. Find the value of $$ab+bc+ca$$ and find at least one triplet $(a,b,c)$ that satisfy those conditions.

Given positive real numbers $a, b, c, d$ such that $cd=1$. Prove that there exists at least one positive integer $m$ such that $$ab\le m^2\le (a+c) (b+d). $$

Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$. (i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$. (ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.

The sequence $(a_n)$ of real numbers is defined as follows: \[a_1=1, \qquad a_2=2, \quad \text{and} \quad a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3.\] Prove that for $n \geq 3$, $a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1$, where $[x]$ denotes the integer $p$ such that $p \leq x < p + 1$.

$\text{(i)}$ Does there exist a positive integer $m > 2016^{2016}$ such that $\frac{2016^m-m^{2016}}{m+2016}$ is a positive integer? $\text{(ii)}$ Does there exist a positive integer $m > 2017^{2017}$ such that $\frac{2017^m-m^{2017}}{m+2017}$ is a positive integer? [i](Serbia MO 2016 P1)[/i]

Set $A$ consists of natural numbers such that these numbers can be expressed as $2x^2+3y^2,$ where $x$ and $y$ are integers. $(x^2+y^2\not=0)$ $a)$ Prove that there is no perfect square in the set $A.$ $b)$ Prove that multiple of odd number of elements of the set $A$ cannot be a perfect square.

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

Let $A$ and $B$ be two $n \times n$ square arrays. The cells of $A$ are labelled by the numbers from $1$ to $n^2$ from left to right starting from the top row, whereas the cells of $B$ are labelled by the numbers from $1$ to $n^2$ along rising north-easterly diagonals starting with the upper left-hand corner. Stack the array $B$ on top of the array $A$. If two overlapping cells have the same number, they are coloured red. Determine those $n$ for which there is at least one red cell other than the cells at top left corner, bottom right corner and the centre (when $n$ is odd). Below shows the arrays for $n=4$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/cc8a435cb28420ccf91340023d440e39f0e849.png[/img]

In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.

Let $\triangle ABC$ be a scalene and acute triangle in which the angle at $A$ is second largest, $H$ is the orthocenter, and $k$ is the circumcircle with center $O$. Let the circumcircle of $\triangle AHO$ intersect the sides $AB$ and $AC$ again at $M$ and $N$, respectively, whereas the altitudes $CH$ and $BH$ intersect $k$ again at $K$ and $L$, respectively. Prove that the intersection of $KL$ and the perpendicular bisector of $AH$ is the orthocenter of $\triangle AMN$. Proposed by [i]Ilija Jovcevski[/i]

For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold: (1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$, (2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and (3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.

At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules: (a) They do not clink glasses crosswise. (b) At each point of time, everyone can clink glasses with at most one other person. How many seconds pass at least until everyone clinked glasses with everybody else? [i](Swiss Mathematical Olympiad 2011, Final round, problem 1)[/i]

The diagonals of a convex $18$-gon are colored in $5$ different colors, each color appearing on an equal number of diagonals. The diagonals of one color are numbered $1, 2,\cdots$. One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly $n$ pairs of diagonals of the same color and with fixed indices $i, j$.

Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and \[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$. Prove that they are converges and find their limits.

Let $x_1 = 97$, and for $n > 1$ let $x_n = \frac{n}{x_{n - 1}}$. Calculate the product $x_1 x_2 \dotsm x_8$.

Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.

Let $n=(q + 2)q^{2021}$ where $q=10^9+7$. For every $k<=n$ and prime $p|n$, define $f_{p,k}(n)$ =$ v_{p}$$ (\binom{n}{k}) $ ($v_{p}$$(i)$ is the highest power of $p$ that divides $i$). Let $m$ be the maximum possible (over all $k$) value of the expression $\prod_{p\text{,prime,} p|n} f_{p,k}$. Find the sum of the digits of $m$.

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

A picture $ 3$ feet across is hung in the center of a wall that is $ 19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? \[ \textbf{(A)}\ 1 \frac{1}{2} \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \frac{1}{2} \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 22 \]

Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments of length $55$, $45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$.