Prove that the sum of the $ \text{2005-th} $ powers of three pairwise distinct complex numbers is the imaginary unit if their modulus are equal and the sum of these numbers is the imaginary unit.

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

[b]p1.[/b] Let $ABCD$ be a square with side length $1$, $\Gamma_1$ be a circle centered at $B$ with radius 1, $\Gamma_2$ be a circle centered at $D$ with radius $1$, $E$ be a point on the segment $AB$ with $|AE| = x$ $(0 < x \le 1)$, and $\Gamma_3$ be a circle centered at $A$ with radius $|AE|$. $\Gamma_3$ intersects $\Gamma_1$ and $\Gamma_2$ inside the square at $G$ and $F$, respectively. Let region $I$ be the region bounded by the segment $GC$ and the minor arc $GC$ of $\Gamma_1$, and region II be the region bounded by the segment $FG$ and the minor arc $FG$ of $\Gamma_3$, as illustrated in the graph below. Let $r(x)$ be the ratio of the area of region I to the area of region II. (i) Find $r(1)$. Justify your answer. (ii) Find an explicit formula of $r(x)$ in terms of $x$ $(0 < x \le 1)$. Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/0/bd2379a1390a578d78dc7e9f4cde756d5f4723.png[/img] [b]p2.[/b] We call a [i]party [/i] any set of people $V$ . If $v_1 \in V$ knows $v_2 \in V$ in a party, we always assume that $v_2$ also knows $v_1$. For a person $v \in V$ in some party, the degree of v, denoted by $deg\,\,(v)$, is the number of people $v$ knows in the party. (i) Suppose that a party has four people with $V = \{v_1, v_2, v_3, v_4\}$, and that $deg\,\,(v_i) = i$ for $i = 1, 2, 3$ show that $deg\,\,(v_4) = 2$. (ii) Suppose that a party is attended by $n = 4k$ ($k \ge 1$) people with $V = \{v_1, v_2, ..., v_{4k}\}$, and that $deg\,\,(v_i) = i$ for $1 \le i \le n - 1$. Show that $deg\,\,(v_n) = \frac{n}{2}$ . [b]p3.[/b] Let $a, b$ be two real number parameters and consider the function $f(x) =\frac{b + \sin x}{a + \cos x}$. (i) Find an example of $(a, b)$ such that $f(x) \ge 2$ for all real numbers $x$. Justify your answer. (ii) If $a > 1$ and the range of the function $f(x)$ (when x varies over the set of all real numbers) is $[-1, 1]$, find the values of $a$ and $b$. Justify your answer. [b]p4.[/b] Let $f$ be the function that assigns to each positive multiple $x$ of $8$ the number of ways in which $x$ can be written as a difference of squares of positive odd integers. (For example, $f(8) = 1$, because $8 = 3^2 -1^2$, and $f(24) = 2$, because $24 = 5^2 - 1^2 = 7^2 - 5^2$.) (a) Determine with proof the value of $f(120)$. (b) Determine with proof the smallest value $x$ for which $f(x) = 8$. (c) Show that the range of this function is the set of all positive integers. [b]p5.[/b] Consider the binomial coefficients $C_{n,r} ={n \choose r}= \frac{n!}{r!(n - r)!}$, for $n \ge 2$. Prove that $C_{n,r}$ are even, for all $1 \le r \le n - 1$, if and only if $n = 2^m$, for some counting number $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

We say that a positive integer is [i]Noble [/i] when: it is composite, it is not divisible by any prime number greater than $20$ and it is not divisible by any perfect cube greater than $1$. How many different Noble numbers are there?

Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are $50^o$, $60^o$ and $70^o$. Find the angles of the original triangle.

Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$ Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

The diagram below shows a triangle divided into sections by three horizontal lines which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has area $100$, find the area of the entire triangle. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; filldraw((-1,2.5)--(-1,1.75)--(0.5,1.75)--(0,2.5)--cycle,grey); draw((-1,4)--(-2,1)); draw((-1,4)--(2,1)); draw((-2,1)--(2,1)); draw((-1,4)--(-1,1)); draw((-1,4)--(-0.5,2.5)); draw((-0.25,1.75)--(0,1)); draw((-1,2.5)--(-1,1.75)); draw((-1,1.75)--(0.5,1.75)); draw((0.5,1.75)--(0,2.5)); draw((0,2.5)--(-1,2.5)); draw((-1.25,3.25)--(-0.25,3.25)); draw((-1.5,2.5)--(0.5,2.5)); draw((1.25,1.75)--(-1.75,1.75)); draw((-1,4)--(0,2.5)); draw((0.47,1.79)--(1,1)); dot((-1,1),dotstyle); dot((0,1),dotstyle); dot((1,1),dotstyle); dot((-1.25,3.25),dotstyle); dot((-1.5,2.5),dotstyle); dot((-1.75,1.75),dotstyle); dot((1.25,1.75),dotstyle); dot((0.5,2.5),dotstyle); dot((-0.25,3.25),dotstyle); [/asy]

Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.

An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.

Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$. Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$.

[b]p1.[/b] The following figure represents a rectangular piece of paper $ABCD$ whose dimensions are $4$ inches by $3$ inches. When the paper is folded along the line segment $EF$, the corners $A$ and $C$ coincide. (a) Find the length of segment $EF$. (b) Extend $AD$ and $EF$ so they meet at $G$. Find the area of the triangle $\vartriangle AEG$. [img]https://cdn.artofproblemsolving.com/attachments/d/4/e8844fd37b3b8163f62fcda1300c8d63221f51.png[/img] [b]p2.[/b] (a) Let $p$ be a prime number. If $a, b, c$, and $d$ are distinct integers such that the equation $(x -a)(x - b)(x - c)(x - d) - p^2 = 0$ has an integer solution $r$, show that $(r - a) + (r - b) + (r - c) + (r - d) = 0$. (b) Show that $r$ must be a double root of the equation $(x - a)(x - b)(x - c)(x - d) - p^2 = 0$. [b]p3.[/b] If $\sin x + \sin y + \sin z = 0$ and $\cos x + \cos y + \cos z = 0$, prove the following statements. (a) $\cos (x - y) = -\frac12$ (b) $\cos (\theta - x) + \cos(\theta - y) + \cos (\theta - z) = 0$, for any angle $\theta$. (c) $\sin^2 x + \sin^2 y + \sin^2 z =\frac32$ [b]p4.[/b] Let $|A|$ denote the number of elements in the set $A$. (a) Construct an infinite collection $\{A_i\}$ of infinite subsets of the set of natural numbers such that $|A_i \cap A_j | = 0$ for $i \ne j$. (b) Construct an infinite collection $\{B_i\}$ of infinite subsets of the set of natural numbers such that $|B_i \cap B_j |$ gives a distinct integer for every pair of $i$ and $j$, $i \ne j$. [b]p5.[/b] Consider the equation $x^4 + y^4 = z^5$. (a) Show that the equation has a solution where $x, y$, and $z$ are positive integers. (b) Show that the equation has infinitely many solutions where $x, y$, and $z$ are positive integers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation. (Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the [i]repetend[/i] of the fraction, and the smallest length of the repetend, $k$, is called the [i]period[/i] of the decimal number.)

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.