For which positive integer couples $(k,n)$, the equality $\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$ holds?

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

Define a sequence $\left\{a_{i}\right\}$ by $a_{1}=3$ and $a_{i+1}=3^{a_{i}}$ for $i \geq 1.$ Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{i} ?$

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .

Is there a function $f(x)$, which satisfies both of the following conditions: a) if $x \ne y$, then $f(x)\ne f(y)$ b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?

Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that $$\left|\alpha-\frac mn\right|<\frac cn.$$

Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$.

Let $n$ be a positive integer and $p_1, p_2, p_3, \ldots p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2$. prove that $6$ divides $n$.

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

For every positive integer $n$ let $$f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}$$ Given $$\sum_{n = 2}^{20} f(n) = \frac{a}{b}$$ for relatively prime positive integers $a$ and $b$, find the sum of the prime factors of $b$. [i]Proposed by Harry Kim[/i]

Suppose $a,b,c$ are positive integers such that \[\gcd(a,b)+\gcd(a,c)+\gcd(b,c)=b+c+2023\] Prove that $\gcd(b,c)=2023$. [i]Remark.[/i] For positive integers $x$ and $y$, $\gcd(x,y)$ denotes their greatest common divisor. [i]Ivan Novak[/i]

In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ?

In a non-isosceles acute triangle $ABC$, $D$ is the midpoint of the edge $[BC]$. The points $E$ and $F$ lie on $[AC]$ and $[AB]$, respectively, and the circumcircles of $CDE$ and $AEF$ intersect in $P$ on $[AD]$. The angle bisector from $P$ in triangle $EFP$ intersects $EF$ in $Q$. Prove that the tangent line to the circumcirle of $AQP$ at $A$ is perpendicular to $BC$.

Consider a solid hemisphere of radius $1$. Find the distance from its center of mass to the base.

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

In the number puzzle below, each cell contains a digit, each cell in the same bolded region has the same digit, and cells in different bolded regions have different digits. The answers to the clues are to be read as three-, four-, or five-digit numbers. Find the unique solution to the puzzle, given that no answer to any clue has a leading $0$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/23514673819aea46c30fd2947f8c82710a1fb3.png[/img]

Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if: a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$. Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown

Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \dots, k$. It is known that for $i = 1, 2, \dots, k$, Part $i$ contains $i$ multiple choice questions, each of which has $(i+1)$ answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly one question correct is equal to $2018$ times the probability that she gets no questions correct. Compute the number of questions that are on the exam. [i]Proposed by Yannick Yao[/i]

Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$, define an [i]$A$-inversion[/i] of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds: [list] [*]$a_i \ge w_i > w_j$ [*]$w_j > a_i \ge w_i$, or [*]$w_i > w_j > a_i$. [/list] Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$, and for any positive integer $k$, the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$-inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$-inversions.

Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N); draw((0,0)--(1,sqrt(3))--(2,0)--(3,sqrt(3))--(4,0)--(5,sqrt(3))--(6,0)); draw((1,0)--(2,sqrt(3))--(3,0)--(4,sqrt(3))--(5,0)); draw((-1.5,0)--(7.5,0)); [/asy] $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 10\sqrt{3}\qquad\textbf{(E)}\ 12\sqrt{3} $

Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

Suppose there exists a group with exactly $n$ subgroups of index 2. Prove that there exists a finite abelian group $G$ that has exactly $n$ subgroups of index 2.