Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector? (Problem from Leningrad)

Points $B,C$ vary on two fixed rays emanating from point $A$ such that $AB+AC$ is constant. Show that there is a point $D$, other than $A$, such that the circumcircle of triangle $ABC$ passes through $D$ for all possible choices of $B, C$.

For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2y}$. The sum of the solutions to the equation \[4096f(f(x, x), x) = x^{13}\] can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.

Let $l$ be a line and $A$ be a point such that $A$ is not on $l.$ Let $P$ be a point on $l$ such that segment $AP$ and line $l$ for a $60^{\circ}$ angle and $AP=1.$ Extend segment $AP$ past $P$ to a point $B$ on the other side of $l.$ Then, let the perpendicular from $B$ to $l$ have foot $M,$ and extend $BM$ past $M$ to $C.$ Finally, extend $CP$ past $P$ to $D.$ Given that $\frac{BP}{AP}=\frac{CM}{BM}=\frac{DP}{CP}=2,$ determine the are of triangle $BPD.$

There are some boys and girls that study in a school. A group of boys is called [i]sociable[/i], if each girl knows at least one of the boys in the group. A group of girls is called [i]sociable[/i], if each boy knows at least one of the girls in the group. If the number of [i]sociable[/i] groups of boys is odd, prove that the number of [i]sociable[/i] groups of girls is also odd.

A [i]palindrome[/i] is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome). Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

Find all prime numbers $p$ for which $x^4\equiv -1\, (mod\, p)$ has a solution.

Assume the quartic $x^4-ax^3+bx^2-ax+d=0$ has four real roots $\frac{1}{2}\leq x_1,x_2,x_3,x_4\leq 2.$ Find the maximum possible value of $\frac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}.$

For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying (1)$0=f(0)<f(1)<f(2)<\ldots$; (2)$f(n)$ has a finite limit when $n$ approaches infinity; (3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.

Let $p$ be a prime. Prove that $p^2+p+1$ is never a perfect cube.

The vertices of $\triangle ABC$ are labeled in counter-clockwise order, and its sides have lengths $CA = 2022$, $AB = 2023$, and $BC = 2024$. Rotate $B$ $90^\circ$ counter-clockwise about $A$ to get a point $B'$. Let $D$ be the orthogonal projection of $B'$ unto line $AC$, and let $M$ be the midpoint of line segment $BB'$. Then ray $BM$ intersects the circumcircle of $\triangle CDM$ at a point $N \neq M$. Compute $MN$. [i]Proposed by Thomas Lam[/i]

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

Let S be the set of positive integers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two rational numbers of the form $\frac{1}{k}$, where $k$ is a positive integer. Prove that $S$ cannot be written as the union of finitely many arithmetic progressions.

Prove that there exists a natural number $b$ such that for any natural $n>b$ the sum of the digits of $n!$ is not less than $10^{100}$. [i]Proposed by D. Khramtsov[/i]

Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.

Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$. (i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$. (ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$. (iii) $f(1)=1$.

Let $CM$ be the median of an acute-angled triangle $ABC$, and $P$ be the projection of the orthocenter $H$ to the bisector of $\angle C$. Prove that $MP$ bisects the segment $CH$.

It is a fact that every set of 2016 consecutive integers can be partitioned in two sets with the following four properties: (i) The sets have the same number of elements. (ii) The sums of the elements of the sets are equal. (iii) The sums of the squares of the elements of the sets are equal. (iv) The sums of the cubes of the elements of the sets are equal. Let $S =\{n + 1; n + 2;$ [b]. . .[/b] $; n + k\}$ be a set of $k$ consecutive integers. (a) Determine the smallest value of $k$ such that property (i) holds for $S$. (b) Determine the smallest value of $k$ such that properties (i) and (ii) hold for $S$. (c) Show that properties (i), (ii) and (iii) hold for $S$ when $k = 8$. (d) Show that properties (i), (ii), (iii) and (iv) hold for $S$ when $k = 16$.

Consider $37$ distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates.

Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them. [i]Proposed by [b]vvluo[/b] & [b]richy[/b][/i]

Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM, CDP, BCN, ADQ$, the first two outward and the other two inward. Prove that $MN = AC$. What can be said about the quadrilateral $MNPQ$?

Suppose $a, b, c$ are real numbers such that $abc \ne 0$. Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$. Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.

Consider a line segment $[AB]$ with midpoint $M$ and perpendicular bisector $m$. For each point$ X \ne M$ on m consider we are the intersection point $Y$ of the line $BX$ with the bisector from the angle $\angle BAX$. As $X$ approaches $M$, then approaches $Y$ to a point of $[AB]$. Which? [img]https://cdn.artofproblemsolving.com/attachments/a/3/17d72a23011a9ec22deb20184717cc9c020a2b.png[/img] [hide=original wording]Beschouw een lijnstuk [AB] met midden M en middelloodlijn m. Voor elk punt X 6= M op m beschouwenwe het snijpunt Y van de rechte BX met de bissectrice van de hoek < BAX . Als X tot M nadert, dan nadert Y tot een punt van [AB]. Welk? [/hide]

Let $n$ be a positive integer. - Find, in terms of $n$, the number of pairs $(x,y)$ of positive integers that are solutions of the equation : $$x^2-y^2=10^2.30^{2n}$$ - Prove further that this number is never a square

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \] forms an arithmetic progression. Prove that the terms of the sequence are equal.