Given two fixed points $A$ and $B$ .The point $M$ runs along the circumference containing $A$ and $B$. $K$ is the midpoint of the segment $[MB]$. $[KP]$ is a perpendicular to the line $(MA)$. a) Prove that all the possible lines $(KP)$ pass through one point. b) Find the set of all the possible points $P$.

Let be a natural number $ n\ge 2 $ and an imaginary number $ z $ having the property that $ |z-1|=|z+1|\cdot\sqrt[n]{2} . $ Denote with $ A,B,C $ the points in the Euclidean plane whose representation in the complex plane are the affixes of $ z,\frac{1-\sqrt[n]{2}}{1+\sqrt[n]{2}} ,\frac{1+\sqrt[n]{2}}{1-\sqrt[n]{2}} , $ respectively. Prove that $ AB $ is perpendicular to $ AC. $

Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$? [i]Proposed by Evin Liang[/i]

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

The notation $\lfloor n \rfloor$ denotes the greatest integer less than or equal to $n$. Evaluate $\lfloor 2.1 \lfloor {-}4.3 \rfloor \rfloor$. $\textbf{(A) }{-}11\qquad\textbf{(B) }{-}10\qquad\textbf{(C) }{-}9\qquad\textbf{(D) }{-}8\qquad\textbf{(E) }{-}4$

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers? $\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}$

Two players take turns alternatively and remove a number from $1,2,\dots,1000$. Players can not remove a number that differ with a number already removed by $1$ also they can not remove a number such that it sums up with another removed number to $1001$. The player who can not move loses. Determine the winner.

Given a convex polyhedron with an even number of edges. Prove that we can attach an arrow to each edge, such that for every vertex of the polyhedron, the number of the arrows ending in this vertex is even.

Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

Prove that if $A,B,$ and $C$ are angles of a triangle measured in radians then $A \cos B +\sin A \cos C >0.$

Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

The exinscribed circle of a triangle $ABC$ corresponding to its vertex $A$ touches the sidelines $AB$ and $AC$ in the points $M$ and $P$, respectively, and touches its side $BC$ in the point $N$. Show that if the midpoint of the segment $MP$ lies on the circumcircle of triangle $ABC$, then the points $O$, $N$, $I$ are collinear, where $I$ is the incenter and $O$ is the circumcenter of triangle $ABC$.

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

Find $[ \sqrt{19992000}]$ where $[a]$ is the greatest integer less than or equal to $x$.

Let three lines forming a triangle $ABC$ be given. Using a two-sided ruler and drawing at most eight lines construct a point $D$ on the side $AB$ such that $\frac{AD}{BD}=\frac{BC}{AC}.$

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

[b]p1.[/b] Suppose $\frac{x}{y} = 0.\overline{ab}$ where $x$ and $y$ are relatively prime positive integers and $ab + a + b + 1$ is a multiple of $12$. Find the sum of all possible values of $y$. [b]p2.[/b] Let $A$ be the set of points $\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}$. How many distinct circles pass through at least three points in $A$? [b]p3.[/b] Jack and Jill need to bring pails of water home. The river is the $x$-axis, Jack is initially at the point $(-5, 3)$, Jill is initially at the point $(6, 1)$, and their home is at the point $(0, h)$ where $h > 0$. If they take the shortest paths home given that each of them must make a stop at the river, they walk exactly the same total distance. What is $h$? [b]p4.[/b] What is the largest perfect square which is not a multiple of $10$ and which remains a perfect square if the ones and tens digits are replaced with zeroes? [b]p5.[/b] In convex polygon $P$, each internal angle measure (in degrees) is a distinct integer. What is the maximum possible number of sides $P$ could have? [b]p6.[/b] How many polynomials $p(x)$ of degree exactly $3$ with real coefficients satisfy $$p(0), p(1), p(2), p(3) \in \{0, 1, 2\}?$$ [b]p7.[/b] Six spheres, each with radius $4$, are resting on the ground. Their centers form a regular hexagon, and adjacent spheres are tangent. A seventh sphere, with radius $13$, rests on top of and is tangent to all six of these spheres. How high above the ground is the center of the seventh sphere? [b]p8.[/b] You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules? [b]p9.[/b] Quadrilateral $ABCD$ has $\overline{AB} = 40$, $\overline{CD} = 10$, $\overline{AD} = \overline{BC}$, $m\angle BAD = 20^o$, and $m \angle ABC = 70^o$. What is the area of quadrilateral $ABCD$? [b]p10.[/b] We say that a permutation $\sigma$ of the set $\{1, 2,..., n\}$ preserves divisibilty if $\sigma (a)$ divides $\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\{1, 2,..., 40\}$ preserve divisibility? (A permutation of $\{1, 2,..., n\}$ is a function $\sigma$ from $\{1, 2,..., n\}$ to itself such that for any $b \in \{1, 2,..., n\}$, there exists some $a \in \{1, 2,..., n\}$ satisfying $\sigma (a) = b$.) [b]p11.[/b] In the diagram shown at right, how many ways are there to remove at least one edge so that some circle with an “A” and some circle with a “B” remain connected? [img]https://cdn.artofproblemsolving.com/attachments/8/7/fde209c63cc23f6d3482009cc6016c7cefc868.png[/img] [b]p12.[/b] Let $S$ be the set of the $125$ points in three-dimension space of the form $(x, y, z)$ where $x$, $y$, and $z$ are integers between $1$ and $5$, inclusive. A family of snakes lives at the point $(1, 1, 1)$, and one day they decide to move to the point $(5, 5, 5)$. Snakes may slither only in increments of $(1,0,0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Given that at least one snake has slithered through each point of $S$ by the time the entire family has reached $(5, 5, 5)$, what is the smallest number of snakes that could be in the family? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How do I prove that, for every $a, b, c$ positive real numbers such that $a+b+c = 1$ the following inequality holds: $\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq \frac{1}{2}$?

Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$, and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$. Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$, respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$, so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$. Prove that $DF,OM,EG$ are concurrent.

Find all triples $(x, y, z)$ of positive integers such that \[x^2 + 4^y = 5^z. \] [i]Proposed by Li4 and ltf0501[/i]

Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that \[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]

The price for sending a packet (a rectangular cuboid) is directly proportional to the sum of its length, width, and height. Is it possible to reduce the cost of sending a packet by putting it into a cheaper packet?

Construct a square from four points, one on each side.

Two masses are connected with spring constant $k$. The masses have magnitudes $m$ and $M$. The center-of-mass of the system is fixed. If $ k = \text {100 N/m} $ and $m=\dfrac{1}{2}M=\text{1 kg}$, let the ground state energy of the system be $E$. If $E$ can be expressed in the form $ a \times 10^p $ eV (electron-volts), find the ordered pair $(a,p)$, where $ 0 < a < 10 $, and it is rounded to the nearest positive integer and $p$ is an integer. For example, $ 4.2 \times 10^7 $ should be expressed as $(4,7)$. [i](Trung Phan, 10 points)[/i]

okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points A[size=67]M[/size], B[size=67]M[/size], C[size=67]M[/size] on the circumcircle of the triangle ABC such that A, A[size=67]M[/size], M are collinear and so on. Find the locus of these points M for which the area of the triangle A[size=67]M[/size] B[size=67]M[/size] C[size=67]M[/size] is bigger than the area of the triangle ABC.