The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$? b) The same question for a square of area $1/2019$. (Mikhail Evdokimov)

Find, with proof, all triples of real numbers $(a, b, c)$ such that all four roots of the polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+b$ are positive integers. (The four roots need not be distinct.)

Find $\alpha$ such that $$\lim_{x\to0^+}x^\alpha I(x)=a\enspace\text{given}\enspace I(x)=\int^\infty_0\sqrt{1+t}\cdot e^{-xt}dt$$ where $a$ is a nonzero real number.

Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Raina Yang[/i]

The angle $\angle XOY =\alpha $ and the points $A$ and $B$ on OY are given such that $OA = a$ and $OB = b$ with $a > b$. A circle passes through the points $A$ and $B$ and is tangent to $OX$. a) Calculate the radius of that circle in terms of $a, b$ and $\alpha $. b) If $a$ and $b$ are constants and $\alpha $ varies, show that the minimum value of the radius of the circle is $\frac{a-b}{2}$.

Solve the equation $p^2-pq-q^3=1$ in prime numbers. [i]A. Golovanov[/i]

Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

(i) Determine the set of all positive integers $n$ for which $3^{n+1}$ divides $2^{3^n} + 1$; (ii) Prove that $3^{n+2}$ does not divide $2^{3^n} + 1$ for any positive integer $n$.

Let a triangle $ABC$ be given. On a ruler three segment congruent to the sides of this triangle are marked. Using this ruler construct the orthocenter of the triangle formed by the tangency points of the sides of $ABC$ with its incircle.

In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. [img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]

The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes? $\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?

A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?