Let $n\geq 1$ be an integer and let $a$ and $b$ be its positive divisors satisfying $a+b+ab=n$. Prove that $a=b$.

Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

Let $n,k$ be given natural numbers. Determine all ordered n-tuples of non-negative real numbers $(x_1,x_2,...,x_n)$ that satisfy the system of equations $$x_1^k+x_2^k+...+x_n^k=1$$ $$(1+x_1)(1+x_2)...(1+x_n)=2$$

Let $A$ and $B$ be nonempty subsets of $\mathbb{N}$. The sum of $2$ distinct elements in $A$ is always an element of $B$. Furthermore, the result of the division of $2$ distinct elements in $B$ (where the larger number is divided by the smaller number) is always a member of $A$. Determine the maximum number of elements in $A \cup B$.

Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. [asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy] Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ $\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113$

In the collection of all right circular cylinders of fixed volume $c$, what is the ratio $\frac hr$ of the cylinder which has the least total surface area?

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

Two positive integers differ by $60.$ The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?

Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$. [i]Proposed by Taiki Aiba[/i]

Is there an infinite periodic sequence of digits for which the following condition condition is fulfilled: for any natural number $n{}$ a natural number divisible by $2^n{}$ can be cut from this sequence of digits (as a word)? [i]Proposed by P. Kozhevnikov[/i]

[color=darkblue]Let $ ABCD$ be a trapezoid with $ AB\parallel CD$. Exterior equilateral triangles $ ABE$ and $ CDF$ are constructed. Prove that lines $ AC$, $ BD$ and $ EF$ are concurrent.[/color]

Let $x, y, z > -1$. Prove that \[ \frac{1+x^2}{1+y+z^2} + \frac{1+y^2}{1+z+x^2} + \frac{1+z^2}{1+x+y^2} \geq 2. \] [i]Laurentiu Panaitopol[/i]

How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?

Use the standard reduction potentials to determine what is observed at the cathode during the electrolysis of a $1.0 \text{M}$ solution of $\ce{KBr}$ that contains phenolphthalein. What observation(s) is(are) made? $\ce{O2 (g)} + \ce{4H^+ (aq)} + 4e^- \rightarrow \ce{2H2O (l) }; \text{ E}^\circ = \text{1.23 V}$ $\ce{Br2 (l)} + 2e^- \rightarrow \ce{2Br^- (aq)} ; \text{ E}^\circ = \text{1.07 V}$ $\ce{2H2O (l)} + 2e^- \rightarrow \ce{H2 (g)} + \ce{2OH^-} ; \text{ E}^\circ = \text{-0.80 V}$ $\ce{K^+ (aq)} + e^- \rightarrow \ce{K (s)} ; \text{ E}^\circ = \text{-2.92 V}$ $ \textbf{(A) }\text{Solid metal forms}\qquad$ $\textbf{(B) }\text{Bubbles form and a pink color appears}\qquad$ $\textbf{(C) }\text{Dark red } \ce{ Br2} \text{ forms}\qquad$ $\textbf{(D) }\text{Bubbles form and the solution remains colorless}\qquad $

A rectangular board, where in each square there is a symbol, is said to be [i]magnificent [/i] if, for each line$ L$ and for each pair of columns $C$ and $D$, there is on the board another line $M$ exactly equal to $L$, except in columns $C$ and $D$, where $M$ has symbols different from those of $L$. What is the smallest possible number of rows on a magnificent board with $2023$ columns?

I was wondering if anyone had a sol for this. I am probably just going to bash it out.

In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are the following: The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$. However, due to an error in the wording of a question, all scores are increased by $5$. At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$. (a) Find all possible values ​​of $N$. (b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.

$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation $$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.

The side lengths of triangle $ABC$ are $13$, $14$, and $15$. Let $I$ be the incenter of the triangle. Compute the product $AI\cdot BI\cdot CI$.

A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $. If the number of photographs taken is huge, find $ k $. That is: what is the time-average of the distance traveled divided by $ h $, dividing by $h$? $ \textbf {(A) } \dfrac{1}{4} \qquad \textbf {(B) } \dfrac{1}{3} \qquad \textbf {(C) } \dfrac{1}{\sqrt{2}} \qquad \textbf {(D) } \dfrac{1}{2} \qquad \textbf {(E) } \dfrac{1}{\sqrt{3}} $ [i]Problem proposed by Ahaan Rungta[/i]

For positive real numbers $x;y;z$ satisfying $0<x,y,z<2$, find the biggest value the following equation could acquire: $$(2x-yz)(2y-zx)(2z-xy)$$

In $\triangle ABC$, let the bisector of $\angle BAC$ hit the circumcircle at $M$. Let $P$ be the intersection of $CM$ and $AB$. Denote by $(V,WX,YZ)$ the intersection of the line passing $V$ perpendicular to $WX$ with the line $YZ$. Prove that the points $(P,AM,AC), (P,AC,AM), (P,BC,MB)$ are collinear. [hide=Restatement]In isosceles triangle $APX$ with $AP=AX$, select a point $M$ on the altitude. $PM$ intersects $AX$ at $C$. The circumcircle of $ACM$ intersects $AP$ at $B$. A line passing through $P$ perpendicular to $BC$ intersects $MB$ at $Z$. Show that $XZ$ is perpendicular to $AP$.[/hide]

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left(x^2+f(y)\right)-y=(f(x+y)-y)^2$$holds for all $x,y\in\mathbb R$.

For every positive integer $m$, a $m\times 2018$ rectangle consists of unit squares (called "cell") is called [i]complete[/i] if the following conditions are met: i. In each cell is written either a "$0$", a "$1$" or nothing; ii. For any binary string $S$ with length $2018$, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied). A [i]complete[/i] rectangle is called [i]minimal[/i], if we remove any of its rows and then making it no longer [i]complete[/i]. a. Prove that for any positive integer $k\le 2018$ there exists a [i]minimal[/i] $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$. b. A [i]minimal[/i] $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$.

Let a triangle $ ABC$. $ E,F$ in segment $ AB$ so that $ E$ lie between $ AF$ and half of circle with diameter $ EF$ is tangent with $ BC,CA$ at $ G,H$. $ HF$ cut $ GE$ at $ S$, $ HE$ cut $ FG$ at $ T$. Prove that $ C$ is midpoint of $ ST$.