In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

Prove that if in a triangle the orthocenter, the centroid and the incenter are collinear, then the triangle is isosceles.

Point $P_1$ is located $600$ miles West of point $P_2$. At $7:00\text{AM}$ a car departs from $P_1$ and drives East at a speed of $50$mph. At $8:00\text{AM}$ another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?

In some school $64$ students participate in five different subject olympiads. In each olympiad at least $19$ students take part; none of them is a participant of more than three olympiads. Prove that if every three olympiads have a common participant, then there are two olympiads having at least five participants in common.

Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$. [i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]

Henry decides one morning to do a workout, and he walks $\tfrac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\tfrac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\tfrac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\tfrac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}$

Functions $f_0(x)=|x|,f_1(x)=|f_0(x)-1|,f_2(x)=|f_1(x)-2|$. Area of the closed part between the figure of $f_2(x)$ and $x$-axis is________.

In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Kendrick Lamar and Drake are cutting their circular beef to share with their fans. The cuts must pass all the way from one side of the beef to the other, and no other modifications may be performed on the beef (e.g. folding, eating, stacking, etc.). Find the minimum number of cuts they will need to split their beef into $2024$ pieces.

In triangle $ABC$, the angles $\alpha=\angle A$ and $\beta=\angle B$ are acute. The isosceles triangle $ACD$ and $BCD$ with the bases $AC$ and $BC$ and $\angle ADC=\beta$, $\angle BEC=\alpha$ are constructed in the exterior of the triangle $ABC$. Let $O$ be the circumcenter of $\triangle ABC$. Prove that $DO+EO$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.

Let $p_1, p_2, \ldots$ be a sequence of primes such that $p_1 =2$ and for $n\geq 1, p_{n+1}$ is the largest prime factor of $p_1 p_2 \ldots p_n +1$ . Prove that $p_n \not= 5$ for any $n$.

[b]4.[/b] Let $\left (H_{\alpha} \right ) $ be a system of sets of integers having the property that for any $\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}$ is a finite set and $H_{{\alpha} _1} \neq H_{{\alpha} _2}$. Prove that there exists a system $\left (H_{\alpha} \right )$ of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets $H_\alpha$ contains more than $K$ elements, then the system $\left (H_{\alpha} \right ) $ is countable ($K$ is an arbitrary fixed integer). [b](St. 4)[/b]

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior? $ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $

On the coordinate plane, draw all points$ M(x, y)$, the coordinates of which satisfy the inequalities $$\cos(x + y)^2 \le \cos(x - y)^2, \,\,\, 0 \le x^3, \,\,\, 0 \le y^3.$$

Find, with proof, all triples of positive integers $(x,y,z)$ satisfying the equation $3^x - 5^y = 4z^2$.

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically? $\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\ $\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\ $\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\ $\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\ $\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ($A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard). [i]V. Chukanov[/i]

Determine the smallest integer $n \ge 1$ such that a $n \times n$ square can be cut into square pieces of size $1 \times 1$ and $2 \times 2$ with both types occuring the same number of times.

Side $AB$ of triangle $ABC$ has length $8$ inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: $\textbf{(A)}\ \dfrac{51}{4} \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ \dfrac{53}{4} \qquad \textbf{(D)}\ \dfrac{40}{3} \qquad \textbf{(E)}\ \dfrac{27}{2} $

Given a positive integer $n$. Prove the inequality $\sum\limits_{i=1}^{n}\frac{1}{i(i+1)(i+2)(i+3)(i+4)}<\frac{1}{96}$

What is the sum $ \frac{1}{1 \cdot 2 \cdot 3} \plus{} \frac{1}{2 \cdot 3 \cdot 4} \plus{} \cdots \plus{} \frac{1}{1996 \cdot 1997 \cdot 1998}$? A. $ \frac{2 \cdot 1997}{3 \cdot 1996 \cdot 1998}$ B. $ \frac{1}{3} \minus{} \frac{1}{3 \cdot 1998}$ C. $ \frac{1}{4} \minus{} \frac{1}{1997^2}$ D. $ \frac{1}{3} \minus{} \frac{1}{3 \cdot 1997 \cdot 1998}$ E. $ \frac{1}{4} \minus{} \frac{1}{2 \cdot 1997 \cdot 1998}$