i) How many integers $n$ are there such that $n$ is divisible by $9$ and $n+1$ is divisible by $25$? ii) How many integers $n$ are there such that $n$ is divisible by $21$ and $n+1$ is divisible by $165$? iii) How many integers $n$ are there such that $n$ is divisible by $9, n + 1$ is divisible by $25$, and $n + 2$ is divisible by $4$?

Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into $6$ parts with the marked areas as in the figure. Show that $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$ [img]https://cdn.artofproblemsolving.com/attachments/a/a/b0a85df58f2994b0975b654df0c342d8dc4d34.png[/img]

Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

Let there be $ 4n \plus{} 2$ distinct paths in space with exactly $ 2n^2 \plus{} 6n \plus{} 1$ points at which exactly two of the paths intersect. (A path never intersects itself.) What is the maximum number of points where exactly three paths intersect?

[b]a)[/b] Show that the solution of the equation $ |z-i|=1 $ in $ \mathbb{C} $ is the set $ \{ 2e^{i\alpha} \sin\alpha |\alpha\in [0,\pi ) \} . $ [b]b)[/b] Let be $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that verify the inequalities $$ \left| z_k-i \right|\le 1,\quad\forall k\in\{ 1,2,\ldots ,n \} . $$ Prove that there exists a complex number $ w $ such that $ |w-i|\le 1 $ and $ w^n=z_1z_2\cdots z_n. $ [i]Dan-Ștefan Marinescu[/i]

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1.[/b] If the values of two angles in a triangle are $60$ and $75$ degrees respectively, what is the measure of the third angle? [b]B2.[/b] Square $ABCD$ has side length $1$. What is the area of triangle $ABC$? [b]B3 / G1.[/b] An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is $8$, what is the square’s side length? [b]B4 / G2.[/b] What is the maximum possible number of sides and diagonals of equal length in a quadrilateral? [b]B5.[/b] A square of side length $4$ is put within a circle such that all $4$ corners lie on the circle. What is the diameter of the circle? [b]B6 / G3.[/b] Patrick is rafting directly across a river $20$ meters across at a speed of $5$ m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of $12$ m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled? [b]B7 / G4.[/b] Quadrilateral $ABCD$ has side lengths $AB = 7$, $BC = 15$, $CD = 20$, and $DA = 24$. It has a diagonal length of $BD = 25$. Find the measure, in degrees, of the sum of angles $ABC$ and $ADC$. [b]B8 / G5.[/b] What is the largest $P$ such that any rectangle inscribed in an equilateral triangle of side length $1$ has a perimeter of at least $P$? [b]G6.[/b] A circle is inscribed in an equilateral triangle with side length $s$. Points $A$,$B$,$C$,$D$,$E$,$F$ lie on the triangle such that line segments $AB$, $CD$, and $EF$ are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon $ABCDEF = \frac{9\sqrt3}{2}$ , find $s$. [b]G7.[/b] Let $\vartriangle ABC$ be such that $\angle A = 105^o$, $\angle B = 45^o$, $\angle C = 30^o$. Let $M$ be the midpoint of $AC$. What is $\angle MBC$? [b]G8.[/b] Points $A$, $B$, and $C$ lie on a circle centered at $O$ with radius $10$. Let the circumcenter of $\vartriangle AOC$ be $P$. If $AB = 16$, find the minimum value of $PB$. [i]The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides. [/i] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).

A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$. [asy] draw((30,0)--(-70,0), Arrow); draw((30,0)--(-20,-40)); draw((-20,-40)--(80,-40), Arrow); draw((0,-60)--(-40,20), dashed, Arrow); draw((0,-60)--(0,15), dashed); draw((0,15)--(40,-65),dashed, Arrow); [/asy]

Find a function $f(x)$ defined for all real values of $x$ such that for all $x$, \[f(x+ 2) - f(x) = x^2 + 2x + 4,\] and if $x \in [0, 2)$, then $f(x) = x^2.$

$(1+x^2)(1-x^3)$ equals $ \text{(A)}\ 1 - x^5\qquad\text{(B)}\ 1 - x^6\qquad\text{(C)}\ 1+ x^2 -x^3\qquad \\ \text{(D)}\ 1+x^2-x^3-x^5\qquad \text{(E)}\ 1+x^2-x^3-x^6 $

$ABCD$ is convex quadrilateral. If $W_a$ is product of power of $A$ about circle $BCD$ and area of triangle $BCD$. And define $W_b,W_c,W_d$ similarly.prove $W_a+W_b+W_c+W_d=0$

Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$

Five spheres of radius $r$ are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone.

A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

Let $ABC$ be an acute triangle with $\angle A = 60$ and altitudes $BE, CF$. Suppose $BE, CF$ are reflected across the perpendicular bisector of $BC$ and the two new segments $B'E', C'F'$ intersect at a point $X$. If $A$ is reflected across $BC$ to form $A'$, show that $AX$ is bisected by the internal angle bisector of $A$.

Let $n\geq3$ be an integer. Find the smallest positive integer $k$ with the property that if in a group of $n$ boys for each boy there are at least $k$ other boys that are born in the same year with him, then all the boys are born in the same year.

Given is a natural number $n > 5$. On a circular strip of paper is written a sequence of zeros and ones. For each sequence $w$ of $n$ zeros and ones we count the number of ways to cut out a fragment from the strip on which is written $w$. It turned out that the largest number $M$ is achieved for the sequence $11 00...0$ ($n-2$ zeros) and the smallest - for the sequence $00...011$ ($n-2$ zeros). Prove that there is another sequence of $n$ zeros and ones that occurs exactly $M$ times.

Two cubes $A$ and $B$ have different side lengths, such that the volume of cube $A$ is numerically equal to the surface area of cube $B$. If the surface area of cube $A$ is numerically equal to six times the side length of cube $B$, what is the ratio of the surface area of cube $A$ to the volume of cube $B$?

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$ $\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$

Let $n$ be even positive integer. There are $n$ real positive numbers written on the blackboard. In every step, we choose two numbers, erase them, and replace [i]each[/i] of then by their product. Show that for any initial $n$-tuple it is possible to obtain $n$ equal numbers on the blackboard after a finite number of steps. [i]Proposed by Peter Novotný[/i]

Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$. Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$, the line parallel to $MN$ through $I$ cuts $AB$, $AC$ in $P$ and $Q$. Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.