Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

Solve the equation $(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$

Let $D$ be a point on the circumcircle of some triangle $ABC$. Let $E, F$ be points on $AC$, $AB$, respectively, such that $A,D,E,F$ are concyclic. Let $M$ be the midpoint of $BC$. Show that if $DM$, $BE$, $CF$ are concurrent, then either $BE \cap CF$ is on the circle $ADEF$, or $EF$ is parallel to $BC$. [i]proposed by USJL[/i]

Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let: \[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \} \] Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.

Positive real numbers $a_1,a_2,\ldots, a_n$ satisfy the equation $$2a_1+a_2+\ldots+a_{n-1}=a_n+\frac{n^2-3n+2}{2}$$ For every positive integer $n \geq 3$ find the smallest possible value of the sum $$\frac{(a_1+1)^2}{a_2}+\ldots+\frac{(a_{n-1}+1)^2}{a_n}$$ [i]M. Zorka[/i]

There are $N$ piles each consisting of a single nut. Two players in turns play the following game. At each move, a player combines two piles that contain coprime numbers of nuts into a new pile. A player who can not make a move, loses. For every $N > 2$ determine which of the players, the first or the second, has a winning strategy.

In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is [asy] size(180); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,1), B=(0,-1), C=(2,-1), D=(2,1), E=(1,-1), F=(1,1), G=(.8,.6); pair X=(4,sqrt(5)), Y=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5)); draw(A--B--C--D--A^^B--F^^E--D^^A--G^^rightanglemark(A,G,F)); draw(X--Y--Z--W--X^^T--V--X^^Y--U); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, S); label("F", F, N); label("G", G, E); label("X", X, NW); label("Y", Y, SW); label("Z", Z, SE); label("W", W, NE); [/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 1+2\sqrt{3}\qquad\textbf{(C)}\ 2\sqrt{5}\qquad\textbf{(D)}\ \frac{8+4\sqrt{3}}{3}\qquad\textbf{(E)}\ 5 $

On the same set of axes are drawn the graph of $ y\equal{}ax^2\plus{}bx\plus{}c$ and the graph of the equation obtained by replacing $ x$ by $ \minus{}x$ in the given equation. If $ b \neq 0$ and $ c \neq 0$ these two graphs intersect: $ \textbf{(A)}\ \text{in two points, one on the x\minus{}axis and one on the y\minus{}axis}\\ \textbf{(B)}\ \text{in one point located on neither axis} \\ \textbf{(C)}\ \text{only at the origin} \\ \textbf{(D)}\ \text{in one point on the x\minus{}axis} \\ \textbf{(E)}\ \text{in one point on the y\minus{}axis}$

Find all complex numbers $m$ such that polynomial \[x^3 + y^3 + z^3 + mxyz\] can be represented as the product of three linear trinomials.

Let $S$ be a $k$-element set. [i](a)[/i] Find the number of mappings $f : S \to S$ such that \[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\] [i](b)[/i] The same with the condition $\text{(i)}$ left out.

For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$

The number of proper subsets of the set $\left\{x|-1\leq\log_{\frac{1}{x}}10<-\frac{1}{2},x\in\mathbb{Z}_{\geq0}\right\}$ is________.

In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.

Let $ABC$ be a triangle such that $M$ and $N$ are the midpoints of $AC$ and $BC$, respectively. Let $I$ be the incenter of $ABC$ and $E$ be the intersection of $MN$ with $Bl$. Let $P$ be a point such that $EP$ is perpendicular to $MN$ and $NP$ parallel to $IA$. Prove that $IP$ is perpendicular to $BC$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Among $18$ parts placed in a row, some three in a row weigh $99 $ g each, and all the rest weigh $100$ g each. On a scale with an arrow, identify all $99$-gram parts.

Let $x, y$, and $z$ be positive integers satisfying the following system of equations: $$x^2 +\frac{2023}{x}= 2y^2$$ $$y +\frac{2028}{y^2} = z^2$$ $$2z +\frac{2025}{z^2} = xy$$ Find $x + y + z$.

How many roots does equation $\sin x = \frac{x}{100}$ have?

Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$. Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar to triangle $ABC$.

In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes on the essay you somehow do not earn any points. It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores? [i]Proposed by Evan Chen[/i]

Let us denote with $S(n)$ the sum of all the digits of $n$. a) Is there such an $n$ that $n+S(n)=1980$? b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.

$$Problem 1 $$The set S = {1,2,3,...,2006} is partitioned into two disjoint subsets A and B such that: (i) 13 ∈ A; (ii) if a ∈ A, b ∈ B, a+b ∈ S, then a+b ∈ B; (iii) if a ∈ A, b ∈ B, ab ∈ S, then ab ∈ A. Determine the number of elements of A

The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?

find all primes $p$, for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$.

Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?