If $a$, $b$, $c > 0$; $ab + bc + ca = 3$ then: $$4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)$$

Let $n$ $\geq$ $2$ and $k$ $\geq$ $2$ be positive integers. A cat and a mouse are playing [i]Wim[/i], which is a stone removal game. The game starts with $n$ stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove $1$, $2$, $\dotsb$, or $k$ stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates [i]Wim 2[/i], which is Wim but with the following additional rule: [i]You cannot remove the same number of stones that your opponent removed on the previous turn[/i]. \\\\Find all values of $k$ such that for every $n$, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

In a country, there are some cities linked together by roads. The roads just meet each other inside the cities. In each city, there is a board which showing the shortest length of the road originating in that city and going through all other cities (the way can go through some cities more than one times and is not necessary to turn back to the originated city). Prove that 2 random numbers in the boards can't be greater or lesser than 1.5 times than each other.

Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$. [i](R. Salimov)[/i]

If the number of triangles that $\angle ABC=60^{\circ},AC=12,BC=k$ is exactly one, then the range value of $k$ is $\text{(A)}k=8\sqrt3\qquad\text{(B)}k=0<k\leq12\qquad\text{(C)}k\geq12\qquad\text{(D)}k=8\sqrt3\text { or }0<k\leq12$

Three distinct integers are chosen uniformly at random from the set $$\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}.$$ Compute the probability that their arithmetic mean is an integer.

The lengths of the sides of triangle $A'B'C'$ are equal to the lengths of the three medians of triangle $ABC.$ Then the ratio $\mathrm{Area} (A'B'C') / \mathrm{Area} (ABC)$ equals $$ \mathrm a. ~ \frac 12\qquad \mathrm b.~\frac 23\qquad \mathrm c. ~\frac34 \qquad \mathrm d. ~\frac56 \qquad \mathrm e. ~\text{Cannot be determined from the information given.} $$

Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.

[b]p1.[/b] Solve the equation $x^2 - x - cos y+1.25 =0$. [b]p2.[/b] Solve the inequality: $\left| \frac{x - 2}{x - 3}\right| \le x$ [b]p3.[/b] Bilbo and Dwalin are seated at a round table of radius $R$. Bilbo places a coin of radius $r$ at the center of the table, then Dwalin places a second coin as near to the table’s center as possible without overlapping the first coin. The process continues with additional coins being placed as near as possible to the center of the table and in contact with as many coins as possible without overlap. The person who places the last coin entirely on the table (no overhang) wins the game. Assume that $R/r$ is an integer. (a) Who wins, Bilbo or Dawalin? Please justify your answer. (b) How many coins are on the table when the game ends? [b]p4.[/b] In the center of a square field is an orc. Four elf guards are on the vertices of that square. The orc can run in the field, the elves only along the sides of the square. Elves run $\$1.5$ times faster than the orc. The orc can kill one elf but cannot fight two of them at the same time. Prove that elves can keep the orc from escaping from the field. [b]p5.[/b] Nine straight roads cross the Mirkwood which is shaped like a square, with an area of $120$ square miles. Each road intersects two opposite sides of the square and divides the Mirkwood into two quadrilaterals of areas $40$ and $80$ square miles. Prove that there exists a point in the Mirkwood which is an intersection of at least three roads. PS. You should use hide for answers.

The sequence $n_1<n_2<\ldots < n_k$ consists of all positive integers $n$ for which in a square $n \times n$ one can mark $10$ cells such that in any square $3 \times 3$ an odd amount of cells are marked. Find $n_{k-2}$.

Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$ [i]Radu Bumbăcea & Călin Popescu[/i]

Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.

For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by \[ W(n,k) = \begin{cases} n^n & k = 0 \\ W(W(n,k-1), k-1) & k > 0. \end{cases} \] Find the last three digits in the decimal representation of $W(555,2)$.

Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.

Let $a, b, c$ be non-negative reals with $a + b + c \le 2$. prove that $$\sqrt{b^2+ac} + \sqrt{a^2+bc} + \sqrt{c^2+ab} \le 3$$

Find all ordered pairs of integers $(x,y)$ such that $5^x = 1 + 4y + y^4$.

Steve has a tricycle which has a front wheel with a radius of $30$ cm and back wheels with radii of $10$ cm and $9$ cm. The axle passing through the centers of the back wheels has a length of $40$ cm and is perpendicular to both planes containing the wheels. Since the tricycle is tilted, it goes in a circle as Steve pedals. Steve rides the tricycle until it reaches its original position, so that all of the wheels do not slip or leave the ground. The tires trace out concentric circles on the ground, and the radius of the circle the front wheel traces is the average of the radii of the other two traced circles. Compute the total number of degrees the front wheel rotates. (Express your answer in simplest radical form.)

Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.

There are 5 points on the plane. The following steps are used to construct lines. In step 1, connect all possible pairs of the points; it is found that no two lines are parallel, nor any two lines perpendicular to each other, also no three lines are concurrent. In step 2, perpendicular lines are drawn from each of the five given points to straight lines connecting any two of the other four points. What is the maximum number of points of intersection formed by the lines drawn in step 2, including the 5 given points?

On a Cartesian plane, $n$ parabolas are drawn, all of which are graphs of quadratic trinomials. No two of them are tangent. They divide the plane into many areas, one of which is above all the parabolas. Prove that the border of this area has no more than $2(n-1)$ corners (i.e. the intersections of a pair of parabolas).

Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

Let $ABC$ be triangle with incenter $I$ . Let $AI$ intersect $BC$ at $D$. Point $P,Q$ lies inside triangle $ABC$ such that $\angle BPA + \angle CQA = 180^\circ$ and $B,Q,I,P,C$ concyclic in order . $BP$ intersect $CQ$ at $X$. Prove that the intersection of $(ABC)$ and $(APQ)$ lies on line $XD$.

Two circles $ O, O'$ having same radius meet at two points, $ A,B (A \not = B) $. Point $ P,Q $ are each on circle $ O $ and $ O' $ $(P \not = A,B ~ Q\not = A,B )$. Select the point $ R $ such that $ PAQR $ is a parallelogram. Assume that $ B, R, P, Q $ is cyclic. Now prove that $ PQ = OO' $.

Consider the equation $x^4-24x^3+210x^2+mx+n=0$. Given that the roots of this equation are nonnegative reals, find the maximum possible value of a root of this equation across all values of $m$ and $n$. [i]Proposed by Andrew Zhao[/i]

Given a circumcircle $(O)$ and two fixed points $B,C$ on $(O)$. $BC$ is not the diameter of $(O)$. A point $A$ varies on $(O)$ such that $ABC$ is an acute triangle. $E,F$ is the foot of the altitude from $B,C$ respectively of $ABC$. $(I)$ is a variable circumcircle going through $E$ and $F$ with center $I$. a) Assume that $(I)$ touches $BC$ at $D$. Probe that $\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}$. b) Assume $(I)$ intersects $BC$ at $M$ and $N$. Let $H$ be the orthocenter and $P,Q$ be the intersections of $(I)$ and $(HBC)$. The circumcircle $(K)$ going through $P,Q$ and touches $(O)$ at $T$ ($T$ is on the same side with $A$ wrt $PQ$). Prove that the interior angle bisector of $\angle{MTN}$ passes through a fixed point.