Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$ Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$ Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct? [i]Proposed by Ivan Chan Kai Chin[/i]

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

A non-self intersecting hexagon $RANDOM$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points $$(0,0), (10,0), (10,10), (0,10), (3,4), (6,2).$$ Let $a_{\text{max}}$ be the greatest possible area of $RANDOM$ and $a_{\text{min}}$ the least possible area of $RANDOM.$ Find $a_{\text{max}}-a_{\text{min}}.$

Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.

Inscribed Triangle $ABC$ on circle $\odot O$. Bisector of $\angle ABC$ intersects $\odot O$ at $D$. Two lines $PB$ and $PC$ that are tangent to $\odot O$ intersect at $P$. $PD$ intersects $AC$ at $E$, $\odot O$ at $F$. $M$ is the midpoint of $BC$. Prove that $M,F,C,E$ are concyclic.

One of the following numbers is not divisible by any prime number less than 10. Which is it? (A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $

Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds: \[ (x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x). \]

$\frac{1}{2^{1990}}(1-3\text{C}_{1990}^2+3^2\text{C}_{1990}^4-3^3\text{C}_{1990}^6+\cdots+3^{994}\text{C}_{1990}^{1988}-3^{995}\text{C}_{1990}^{1990})=$________.

Which of the following is true if $S = \dfrac 1{1^2} + \dfrac 1{2^2} + \dfrac 1{3^2} + \cdots + \dfrac 1{2001^2} + \dfrac 1{2002^2}$? $ \textbf{a)}\ 1\leq S < \dfrac 43 \qquad\textbf{b)}\ \dfrac 43 \leq S < 2 \qquad\textbf{c)}\ 2 \leq S < \dfrac 73$ $\textbf{d)}\ \dfrac 73 \leq S < \dfrac 52 \qquad\textbf{e)}\ \dfrac 52 \leq S < 3 $

$a,b$ are real numbers such that: \[ a^3+b^3=8-6ab. \] Find the maximal and minimal value of $a+b$.

Let $f:\mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ be defined by $f(0)=0$, $$f(2n+1)=2f(n)$$ for $n \ge 0$ and $$f(2n)=2f(n)+1$$ for $n \ge 1$ If $g(n)=f(f(n))$, prove that $g(n-g(n))=0$ for all $n \ge 0$.

If $ a\otimes b =\frac{a+b}{a-b} $ , then $ (6\otimes 4)\otimes 3 = $ = $ \text{(A)}\ 4\qquad\text{(B)}\ 13\qquad\text{(C)}\ 15\qquad\text{(D)}\ 30\qquad\text{(E)}\ 72 $

$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

Let be four distinct complex numbers $ a,b,c,d $ chosen such that $$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$ and $$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$ Calculate $ |a-c| $ and $ |a-d|. $ [i]Carmen Botea[/i]

Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second? [i](3 points)[/i]

In a lecture hall some chairs are placed in rows and columns, forming a rectangle. In each row there are $6$ boys sitting and in each column there are $8$ girls sitting, whereas $15$ places are not taken. What can be said about the number of rows and that of columns?

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.

Incircle of $ABC$ tangent $AB,AC,BC$ in $C_1,B_1,A_1$. $AA_1$ intersect incircle in $E$. $N$ is midpoint $B_1A_1$. $M$ is symmetric to $N$ relatively $AA_1$. Prove that $\angle EMC= 90$

A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?

Two teams of movers are lowering a piano from the window of a $10$ floor apartment building. The rope breaks when the piano is $30$ meters above the ground. The movers on the ground, alerted by the shouts of the movers above, first notice the piano when it is $14$ meters above the ground. How long do they have to get out of the way before the piano hits the ground? (A) $\text{0.66 sec}$ (B) $\text{0.78 sec}$ (C) $\text{1.67 sec}$ (D) $\text{1.79 sec}$ (E) $\text{2.45 sec}$

In a cube with side length $6$, what is the volume of the tetrahedron formed by any vertex and the three vertices connected to that vertex by edges of the cube?

In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. [img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]

Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote \[ S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|. \] a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series \[ \sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c) \] are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent? b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series \[ S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d) \] are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?

A triangle $XY Z$ and a circle $\omega$ of radius $2$ are given in a plane, such that $\omega$ intersects segment $\overline{XY}$ at the points $A$, $B$, segment $\overline{Y Z}$ at the points $C$, $D$, and segment $\overline{ZX}$ at the points $E$, $F$. Suppose that $XB > XA$, $Y D > Y C$, and $ZF > ZE$. In addition, $XA = 1$, $Y C = 2$, $ZE = 3$, and $AB = CD = EF$. Compute $AB$.