Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

Let $ M$ be a point in triangle $ ABC$ such that $ \angle AMC\equal{}90^{\circ}$, $ \angle AMB\equal{}150^{\circ}$, $ \angle BMC\equal{}120^{\circ}$. The centers of circumcircles of triangles $ AMC,AMB,BMC$ are $ P,Q,R$, respectively. Prove that the area of $ \triangle PQR$ is greater than the area of $ \triangle ABC$.

Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that \[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]

Present a family of subsets of the plane such that each one of its members is Lebesgue measurable, each one of its members intersects any circle, and the set of Lebesgue measures of all its members is the set of nonnegative real numbers.

Prove the inequality \[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \] where $a_i > 0, i = 1, 2, 3, 4.$

A soccer ball is usually made from a polyhedral fugure model, with two types of faces, hexagons and pentagons, and in every vertex incide three faces - two hexagons and one pentagon. We call a polyhedron [i]soccer-ball[/i] if it is similar to the traditional soccer ball, in the following sense: its faces are \(m\)-gons or \(n\)-gons, \(m \not= n\), and in every vertex incide three faces, two of them being \(m\)-gons and the other one being an \(n\)-gon. [list='i'] [*] Show that \(m\) needs to be even. [*] Find all soccer-ball polyhedra. [/list]

Line segment $AM=d>0$ is given in the plane. Furthermore, a positive number $v$ is given. Construct a right triangle $ABC$ with hypotenuse $AB$, altitude to the hypotenuse of the length $v$ and the leg $BC$ being divided by $M$ in ration $MB/MC=2/3$. Discuss conditions of solvability in terms of $d, v$.

The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is: $ \textbf{(A) }\text{a point}\qquad\textbf{(B) }\text{ a straight line} \qquad\textbf{(C) }\text{two straight lines}\qquad\textbf{(D) }\text{a circle}\qquad$ $\textbf{(E) }\text{two circles} $

Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Determine, with proof, whether $22!6! + 1$ is prime.

A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]

Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of $100 \dfrac{\text{ft}}{\text{s}}$. At the same time, Edward started running away with an acceleration of $1 \dfrac{\text{ft}}{\text{s}^2}$. Assuming that air resistance causes the arrow to decelerate at $1 \dfrac{\text{ft}}{\text{s}^2}$, and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\dfrac{\text{ft}}{\text{s}}$)?

For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.

Let $n$ be an integer greater than $1$. In the Cartesian coordinate system we consider all squares with integer vertices $(x,y)$ such that $1\le x,y\le n$. Denote by $p_k\ (k=0,1,2,\ldots )$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\sum_k(k-1)p_k=0$.

Let $M$ be the centroid of $\Delta ABC$ Prove the inequality $\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$  (a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$ (b) for any $\Delta ABC$