$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.

a) The triangle $ABC$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$. b) The quadrangle $ABCD$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B_1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram.

Let $n$ be a positive integer and let $t$ be an integer. $n$ distinct integers are written on a table. Bob, sitting in a room nearby, wants to know whether there exist some of these numbers such that their sum is equal to $t$. Alice is standing in front of the table and she wants to help him. At the beginning, she tells him only the initial sum of all numbers on the table. After that, in every move he says one of the $4$ sentences: $i.$ Is there a number on the table equal to $k$? $ii.$ If a number $k$ exists on the table, erase him. $iii.$ If a number $k$ does not exist on the table, add him. $iv.$ Do the numbers written on the table can be arranged in two sets with equal sum of elements? On these questions Alice answers yes or no, and the operations he says to her she does (if it is possible) and does not tell him did she do it. Prove that in less than $3n$ moves, Bob can find out whether there exist numbers initially written on the board such that their sum is equal to $t$

Let $f(n)=\frac{n}{3}$ if $n$ is divisible by $3$ and $f(n)=4n-10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c)=2$. (Here $f^5(x)$ means $f(f(f(f(f(x)))))$.) [i]Proposed by Justin Stevens[/i]

A $9\times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$. We will call [i]special [/i] rectangle any rectangle of $2\times 3$ or $3\times 2$ or $4\times 5$ or $5\times 4$ on the board. The permitted operations are: $\bullet$ Simultaneously add $1$ to all the numbers located in a special rectangle. $\bullet$ Simultaneously subtract $1$ from all numbers located in a special rectangle. Demonstrate that it is possible to achieve, through a succession of permitted operations, that $80$ squares to have $0$ (zero). What number is left in the remaining box?

Let $n$ be a natural number. To prove that the value of the expression $$\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)}$$ does not depend on the choice of the different real numbers $x_0, x_1, ... , x_n$.

Let $AB$ be a chord of a circle $\Gamma$ and let $C$ be a point on the segment $AB$. Let $r$ be a line through $C$ which intersects $\Gamma$ at the points $D,E$; suppose that $D,E$ lie on different sides with respect to the perpendicular bisector of $AB$. Let $\Gamma_D$ be the circumference which is externally tangent to $\Gamma$ at $D$ and touches the line $AB$ at $F$. Let $\Gamma_E$ be the circumference which is externally tangent to $\Gamma$ at $E$ and touches the line $AB$ at $G$. Prove that $CA=CB$ if and only if $CF=CG$.

Given $n(\geq 3)$ distinct points in the plane, no three of which are on the same straight line, prove that there exists a simple closed polygon with these points as vertices.

Determine the number of ordered pairs of positive integers $(x,y)$ with $y < x \le 100$ such that $x^2-y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are [i]relatively prime[/i] if they have no common factor other than $1$.) [i]Ray Li[/i]

Find, with proof, the smallest real number $C$ with the following property: For every infinite sequence $\{x_i\}$ of positive real numbers such that $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$ for $n = 1, 2, 3, \cdots$, we have \[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

Points $X,Y,Z$ lies on a line (in indicated order). Triangles $XAB$, $YBC$, $ZCD$ are regular, the vertices of the first and the third triangle are oriented counterclockwise and the vertices of the second are opposite oriented. Prove that $AC$, $BD$ and $XY$ concur. V.A.Yasinsky

Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$, respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$.

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.

Each of the thirty squares in the diagram below contains a number $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to $S$. [img]https://cdn.artofproblemsolving.com/attachments/8/0/3e056ebc252aee9ade1f45fd337cc6a2f84302.png[/img]

A [i]fortress[/i] is a finite collection of cells in an infinite square grid with the property that one can pass from any cell of the fortress to any other by a sequence of moves to a cell with a common boundary line (but it can have "holes"). The [i]walls[/i] of a fortress are the unit segments between cells belonging to the fortress and cells not belonging to the fortress. The [i]area[/i] $A$ of a fortress is the number of cells it consists of. The [i]perimeter[/i] $P$ is the total length of its walls. Each cell of the fortress can contain a [i]guard[/i] which can oversee the cells to the top, the bottom, the right and the left of this cell, up until the next wall (it also oversees its own cell). (a) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of perimeter $P \le 2024$. (b) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of area $A \le 2024$.

To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.

How many positive integers $ n$ satisfy the following condition: \[ (130n)^{50} > n^{100} > 2^{200}? \]$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 65\qquad \textbf{(E)}\ 125$

For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$

Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.

Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that \[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]

Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.

Let $a,b,c$ be integers larger than $1$. Prove that \[a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.\]

23) A spring has a length of 1.0 meter when there is no tension on it. The spring is then stretched between two points 10 meters apart. A wave pulse travels between the two end points in the spring in a time of 1.0 seconds. The spring is now stretched between two points that are 20 meters apart. The new time it takes for a wave pulse to travel between the ends of the spring is closest to which of the following? A) 0.5 seconds B) 0.7 seconds C) 1 second D) 1.4 seconds E) 2 seconds