Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9:00$ the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance ?

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has zero, one or more than one real solutions if $(b-1)^2-4ac$ is negative, equal to zero or positive respectively.

Click for solution Consider the function \[ f(x)=\left(\frac{2ax+b-1+\sqrt{\Delta}}{2ax+b-1-\sqrt{\Delta}}\right)^{\frac{1}{\sqrt{\Delta}}} \] where $\Delta=(b-1)^2-4ac$. If $x_i$ is an approximation to a root of this function, then $x_{i+1}$ is next approximation to the root of this function obtained by using the Newton-Raphson method. So a solution of the system of equations corresponds to a failure of the Newton-Raphson method to find a root of this function, producing instead a cycle of length $n$.

Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality \[(n + 1)h_n+1 - nh_n > r.\] Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$

If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation \[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\] has at least $n - 1$ real roots.

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?

Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following: (i) $|AB| \leq a$ ($a$ fixed); (ii) $|EF| = l$ ($l$ fixed); (iii) the sum of squares of the nonparallel sides of the trapezoid is constant. RemarkRemark. The constants are chosen so that such trapezoids exist.

Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. (a) For what values of $a/b$ can these segments be sides of a triangle ? (b) For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation \[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]

If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that \[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]

Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ ($A_iA_k = B_iB_k$), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.

A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$ . Find the locus of the center of the circumcircle of triangle $XAY .$

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] RemarkFor any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.

Click for solution It immediately comes from base 2 representation. Indeed, suppose $n=\overline{a_{n}...a_2a_1a_0}$ (actually, we know that it containes at most $n-1$ digit for $n\geq 1$). And we have $\left[\frac{n+2^{k-1}}{2^{k}}\right]=\left[\frac{n}{2^{k}}+\frac{1}{2}\right]=\overline{a_{n-1}...a_{k}}+a_{k-1}$ . Therefore the whole sum is $\sum a_k(2^k-1)+\sum a_{k-1}= \sum a_k2^k=n$.

A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$

Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x, OB = y, OC = z$, if and only if $x+y \geq z, y+z \geq x, z+x \geq y$ (the points $O,A,B,C$ are coplanar).

If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ . $

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

Find all complex numbers $m$ such that polynomial \[x^3 + y^3 + z^3 + mxyz\] can be represented as the product of three linear trinomials.

Given $k$ parallel lines $l_1, \ldots, l_k$ and $n_i$ points on the line $l_i, i = 1, 2, \ldots, k$, find the maximum possible number of triangles with vertices at these points.

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.