Prove that there is no inclined plane such that any tetrahedron placed arbitrarily with a certain face on the plane will not fall over. It means the following: Given a plane $ \pi $ and a line $ l $ not perpendicular to it. Prove that there is a tetrahedron $ T $ such that for each of its faces $ S $ there is in the plane $ \pi $ a triangle $ ABC $ congruent to $ S $ and there is a point $ D $ such that the tetrahedron $ ABCD $ congruent to $ T $ and the line parallel to $ l $ passing through the center of gravity of the tetrahedron $ ABCD $ does not intersect the triangle $ ABC $. Note. The center of gravity of a tetrahedron is the intersection point of the segments connecting the centers of gravity of the faces of this tetrahedron with the opposite vertices (it is known that such a point always exists).

In the electoral college, each of $51$ places get some positive number of electoral votes for a nationwide total of $538$. Thus, $270$ electoral votes guarantees a win. Across all distributions of electoral votes to each place, let $M$ be the maximum number of sets of places that combine to have at least $270$ electoral votes. Find $M$.

A line passing through $(20,21)$ intersects the curve $y = x^3-2x^2-3x+5$ at three distinct points $A, B,$ and $C,$ such that $B$ is the midpoint of $\overline{AC}$. The slope of this line is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.

Suppose that real numbers $x,y,z$ satisfy $$\frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=p.$$Prove that $\cos(x+y)+\cos(y+z)+\cos(x+z)=p$.

Find the number of pairs of integers $(a,b)$ with $0 \le a,b \le 2019$ where $ax \equiv b \pmod{2020}$ has exactly $2$ integer solutions $0 \le x \le 2019$. [i]Proposed by Richard Chen[/i]

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

Let $ABC$ be a triangle, and let $D$ and $E$ be two points on side $BC$ such that $BD = EC$. Let $X$ be a point on segment $AD$ such that $CX$ is parallel to the bisector of $\angle ADB$. Similarly, let $Y$ be a point on segment $AD$ such that $BY$ is parallel to the bisector of $\angle ADC$. Prove that $DE = XY$.

In a triangle $ABC$, the bisectors of the angles at $B$ and $A$ meet the opposite sides at $P$ and $Q$, respectively. Suppose that the circumcircle of triangle $PQC$ passes through the incenter $R $ of $\vartriangle ABC$. Given that $PQ = l$, find all sides of triangle $PQR$.

In a class of $30$ students, each students knows exactly six other students. (Of course, knowing is a mutual relation, so if $A$ knows $B$, then $B$ knows $A$). A group of three students is balanced if either all three students know each other, or no one knows anyone else within that group. How many balanced groups exist?

Let $f$ be a function that associates a positive integer $(x, y)$ with each pair of positive integers $f(x, y)$. Suppose that $f(x, y) \le xy$ for all positive integers $x$, $y$. Show that there are $2023$ different pairs $(x_1, y_1)$,$...$, $ (x_{2023}, y_{2023}$) such that $$f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).$$

Prove that $\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ba} \ge a + b + c$, for all positive real numbers $a, b$, and $c$.

Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$\\ For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

There are several contestants at a math olympiad. We say that two contestants $ A$ and $ B$ are [i]indirect friends[/i] if there are contestants $ C_1, C_2, ..., C_n$ such that $ A$ and $ C_1$ are friends, $ C_1$ and $ C_2$ are friends, $ C_2$ and $ C_3$ are friends, ..., $ C_n$ and $ B$ are friends. In particular, if $ A$ and $ B$ are friends themselves, they are [i]indirect friends[/i] as well. Some of the contestants were friends before the olympiad. During the olympiad, some contestants make new friends, so that after the olympiad every contestant has at least one friend among the other contestants. We say that a contestant is [i]special[/i] if, after the olympiad, he has exactly twice as indirect friends as he had before the olympiad. Prove that the number of special contestants is less or equal than $ \frac{2}{3}$ of the total number of contestants.

Find all polynomials $P(x)$ which have the properties: 1) $P(x)$ is not a constant polynomial and is a mononic polynomial. 2) $P(x)$ has all real roots and no duplicate roots. 3) If $P(a)=0$ then $P(a|a|)=0$ [i](nooonui)[/i]

In rectangular coodinate system $Oxy$, consider the line $y=3x$ and point $A(4,3)$. Find on the line $y=3x$, point $B\ne O$ such that the area of triangle $OBC$ is the minimum possible, where $C= AB\cap Ox$.

Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.

If $ a$, $ b$, $ c$, and $ d$ are positive real numbers such that $ a$, $ b$, $ c$, $ d$ form an increasing arithmetic sequence and $ a$, $ b$, $ d$ form a geometric sequence, then $ \frac{a}{d}$ is $ \textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

Let $ABC$ be an acute triangle, and let $F_A$ and $F_B$ be the midpoints of sides $BC$ and $CA$, respectively. Let $E$ and $F$ be the intersection points of the circle centered at $F_A$ and passing through $A$ and the circle centered at $F_B$ and passing through $B$. Prove that if segments $CE$ and $CF$ have midpoints $N$ and $M$, respectively, then the intersection points of the circle centered at $M$ and passing through $E$ and the circle centered at $N$ and passing through $F$ lie on the line $AB$.

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

Let $ABC$ be an acute triangle with $|AB| > |AC|.$ Let $D$ be a point on the side $AB$, such that the angles $\angle ACD$ and $\angle CBD$ are equal. Let $E$ denote the midpoint of $BD,$ and let $S$ be the circumcenter of the triangle $BCD.$ Prove that the points $A, E, S$ and $C$ lie on the same circle.