Let $n$ be a positive integer . If $n$ has odd number of divisors ( other than $1$ and $n$ ) , then show that $n$ is a perfect square .

A surjective function $g: \mathbb{C} \to \mathbb C$ is given. Find all functions $f: \mathbb{C} \to \mathbb C$ such that for all $x,y\in \mathbb C$ we have $$ |f(x)+g(y)| = | f(y) + g(x)|. $$ Proposed by [i]Mojtaba Zare, Amirabbas Mohammadi[/i]

Let $ABC$ be an obtuse triangle with obtuse angle $\angle B$ and altitudes $AD, BE, CF$. Let $T$ and $S$ be the midpoints of $AD$ and $CF$, respectively. Let $M$ and $N$ and be the symmetric images of $T$ with respect to lines $BE$ and $BD$, respectively. Prove that $S$ lies on the circumcircle of triangle $BMN$.

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

Let equilateral triangle $\vartriangle ABC$ be inscribed in a circle $\omega_1$ with radius $4$. Consider another circle $\omega_2$ with radius $2$ internally tangent to $\omega_1$ at $A$. Let $\omega_2$ intersect sides $\overline{AB}$ and $\overline{AC}$ at $D$ and $E$, respectively, as shown in the diagram. Compute the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/a/5/8255cdb8b041719d735607da8139aa4016375d.png[/img]

Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that [list=disc] [*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and [*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$. [/list] Prove that $\max(a_1,a_{2023})\ge 507$.

Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i=1}^4 x_i=98.$ Find $\frac n{100}.$

Let $ U \equal{} 2\times2004^{2005}$, $ V \equal{} 2004^{2005}$, $ W \equal{} 2003\times2004^{2004}$, $ X \equal{} 2\times2004^{2004}$, $ Y \equal{} 2004^{2004}$ and $ Z \equal{} 2004^{2003}$. Which of the following is the largest? $ \textbf{(A)}\ U \minus{} V \qquad \textbf{(B)}\ V \minus{} W \qquad \textbf{(C)}\ W \minus{} X \qquad \textbf{(D)}\ X \minus{} Y \qquad \textbf{(E)}\ Y \minus{} Z$

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

A man buys a house for $10000$ dollars and rents it. He puts $ 12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325$ dollars a year taxes and realizes $ 5\frac{1}{2}\%$ on his investment. The monthly rent is: $\textbf{(A)}\ 64.82\text{ dollars} \qquad \textbf{(B)}\ 83.33\text{ dollars} \qquad \textbf{(C)}\ 72.08\text{ dollars} \qquad \textbf{(D)}\ 45.83\text{ dollars} \qquad \textbf{(E)}\ 177.08\text{ dollars}$

[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y,z$ be positive real numbers satisfying $4x^2 - 2xy + y^2 = 64, y^2 - 3yz +3z^2 = 36,$ and $4x^2 +3z^2 = 49.$ If the maximum possible value of $2xy +yz -4zx$ can be expressed as $\sqrt{n}$ for some positive integer $n,$ find $n.$

On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k\geq 0$ lines. How many different values of $k$ are possible? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

Two vertices of the rectangle are located on side $BC$ of triangle $ABC$, and the other two are on sides $AB$ and $AC$. It is known that the midpoint of the altitude of this triangle, drawn on the side $BC$, lies on one of the diagonals of the rectangle, and the side of the rectangle located on $BC$ is three times less than $BC$. In what ratio does the altitude of the triangle divide the side $BC$ ?

Let $k$ be a natural number. Bijection $f:\mathbb{Z} \rightarrow \mathbb{Z}$ has the following property: for any integers $i$ and $j$, $|i-j|\leq k$ implies $|f(i) - f(j)|\leq k$. Prove that for every $i,j\in \mathbb{Z}$ it stands: \[|f(i)-f(j)|= |i-j|.\]

An infinite array of rational numbers $G(d, n)$ is defined for integers $d$ and $ n$ with $1\leq d \leq n$ as follows: $$G(1, n)= \frac{1}{n}, \;\;\; G(d,n)= \frac{d}{n} \sum_{i=d}^{n} G(d-1, i-1) \; \text{for} \; d>1.$$ For $1 < d < p$ and $p$ prime, prove that $G(d, p)$ is expressible as a quotient $s\slash t$ of integers $s$ and $t$ with $t$ not divisible by $p.$

$ABC$ is a triangle such that $AC<AB<BC$ and $D$ is a point on side $AB$ satisfying $AC=AD$. The circumcircle of $ABC$ meets with the bisector of angle $A$ again at $E$ and meets with $CD$ again at $F$. $K$ is an intersection point of $BC$ and $DE$. Prove that $CK=AC$ is a necessary and sufficient condition for $DK \cdot EF = AC \cdot DF$.

The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$. You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$. (a) Find $g(2021)$. (b) Find $g(1/2021)$.

Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

Consider a triangle $ABC$ with $AB=AC$, and $D$ the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE= \angle ECB=18^{\circ}$. If $AD=3$, find the length of the segment $CE$.

Let $\{a_n\}$ be the sequance with $a_0=0$, $a_n=\frac{1}{a_{n-1}-2}$ ($n\in N_+$). Select an arbitrary term $a_k$ in the sequence $\{a_n\}$ and construct the sequence $\{b_n\}$: $b_0=a_k$, $b_n=\frac{2b_{n-1}+1} {b_{n-1}}$ ($n\in N_+$) . Determine whether the sequence $\{b_n\}$ is a finite sequence or an infinite sequence and give proof.

Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds: $ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $

What is the shortest distance between the plane $Ax + By + Cz + 1 = 0$ and the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$ You may find it convenient to use the notation $h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}.$ What is the algebraic condition for the plane not to intersect the ellipsoid?