There is an equilateral triangle with side $d{}$ and a point $P{}$ such that the distances from $P{}$ to the vertices of the triangle are positive numbers $a, b, c$. Prove that there exist a point $Q{}$ and an equilateral triangle with side $a{}$, such that the distances from $Q{}$ to the vertices of this triangle are $b, c, d$. [i]Alexandr Evnin[/i]

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $ABC$ be a triangle with bisectors $BE$ and $CF$ meet at $I$. Let $D$ be the projection of $I$ on the $BC$. Let M and $N$ be the orthocenters of triangles $AIF$ and $AIE$, respectively. Lines $EM$ and $FN$ meet at $P.$ Let $X$ be the midpoint of $BC$. Let $Y$ be the point lying on the line $AD$ such that $XY \perp IP$. Prove that line $AI$ bisects the segment $XY$. [i]Proposed by Tran Quang Hung - Vietnam[/i]

An $n \times n$ complex matrix $A$ is called \emph{t-normal} if $AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Proposed by Shachar Carmeli, Weizmann Institute of Science

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A\neq Z$). Let $B$ be the centre of $\Gamma_1$ and let $C$ be the centre of $\Gamma_2$. The exterior angle bisector of $\angle{BAC}$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Prove that the interior angle bisector of $\angle{BZC}$ passes through the circumcenter of $\triangle{XYZ}$. [i]For points $P,Q,R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle{PQS}$ is the line that divides $\angle{PQS}$ into two equal angles, while the exterior angle bisector of $\angle{PQS}$ is the line that divides $\angle{RQS}$ into two equal angles.[/i]

The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^2$, $8\text{ in}^2$, and $6\text{ in}^2$ respectively is: $\textbf{(A)}\ 576\text{ in}^3 \qquad \textbf{(B)}\ 24\text{ in}^3 \qquad \textbf{(C)}\ 9\text{ in}^3 \qquad \textbf{(D)}\ 104\text{ in}^3 \qquad \textbf{(E)}\ \text{None of these}$

For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and \[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\ 2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases} \] Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ . Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.

Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line. [i]Proposed by Elliott Liu and Anthony Wang[/i]

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

Find the Betti numbers of the surface $x^2+y^2 = 1 + z^2$ in three-dimensional projective space. The same for the surfaces $z = xy$, $z=x^2$, $z^2 = x^2 + y^2$.

Show that ten equal-sized squares cannot be placed on a plane in such a way that no two have an interior point in common and the first touches each of the others.

A lattice point $(x, y, z) \in Z^3$ can be seen from the origin if the line from the origin does not contain any other lattice point $(x', y', z')$ with $$(x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2.$$ Let $p$ be the probability that a randomly selected point on the cubic lattice $Z^3$ can be seen from the origin. Given that $$\frac{1}{p}= \sum^{\infty}_{n=i} \frac{k}{n^s}$$ for some integers $ i, k$, and $s$, find $i, k$ and $s$.

Find all integers $n$ such that $\frac{4n}{n^2 +3 }$is an integer.

We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$. [i]Solution by Mostafa Eynollahzade and Erfan Salavati[/i]

Real numbers$ x$ and $y$ it is know that $\frac{x+y}{x-y}+\frac{x-y}{x+y}= 1999$. Find the value of the expression $$\frac{x^4+y^4}{x^4-y^4}+\frac{x^4-y^4}{x^4+y^4}.$$

Does there exist a sequence of positive integers \( a_1, a_2, \ldots, a_{100} \) such that every number from \( 1 \) to \( 100 \) appears exactly once, and for each \( 1 \leq i \leq 100 \), the condition \[ a_{a_i + i} = i \] holds? Here it is assumed that \( a_{k+100} = a_k \) for each \( 1 \leq k \leq 100 \). [i]Proposed by Mykhailo Shtandenko[/i]

Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^k>N$. Find the minimum possible value of \[\dfrac{10^k-1}{\gcd(N,10^k-1)}.\]

In one night the air temperature remained constant, several degrees below zero, and that of the water of a very extensive cylindrical pond, which formed a layer $10$ cm deep, it reached zero degrees, beginning then to form a layer of ice on the surface. Under these conditions it can be assumed that the thickness of the ice sheet formed is directly proportional to the square root of the time elapsed. At $0$ h, the thickness of the ice was $3$ cm and at $4$ h it was just over to freeze the water in the pond. Calculate at what time the ice sheet began to form, knowing that the density of the ice formed was $0.9$.

In $\mathbb{R}^3$ some $n$ points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point $P \in \mathbb{R}^3$ in a finite number of steps (possibly depending on $P$). Find the minimal value of $n$ for which this could happen.

You are given a number, and round it to the nearest thousandth, round this result to nearest hundredth, and round this result to the nearest tenth. If the final result is $.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as a decimal.