Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$

Evaluate the continued fraction $$1+\frac{2}{2+\frac{2}{2+\ldots}}$$ [i]2015 CCA Math Bonanza Team Round #4[/i]

There are exactly 7 possible tetrominos (groups of 4 connected squares in a grid): [img]https://cdn.discordapp.com/attachments/813077401265242143/816189385859006474/tetris.png[/img] Daniel has a $2 \times 20210$ rectangle and wants to tile the interior with tetrominos without overlaps, pieces sticking out, or extra pieces left over. Note that you are allowed to rotate tetrominos but not reflect them. For how many multisets of tetrominos (ie. an ordered tuple of how many of each tile he has) is it possible to exactly tile his $2\times20210$ rectangle? [i]Proposed by Dilhan Salgado[/i]

Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

Find all real values of the parameter $a$ for which the system \begin{align*} &1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\ &\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3) \end{align*}has a unique real solution. Solve the system for these values of $a$.

Let $ a,b,c,x,y$ be five real numbers such that $ a^3 \plus{} ax \plus{} y \equal{} 0$, $ b^3 \plus{} bx \plus{} y \equal{} 0$ and $ c^3 \plus{} cx \plus{} y \equal{} 0$. If $ a,b,c$ are all distinct numbers prove that their sum is zero. [i]Ciprus[/i]

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

Show that for every polynomial $W$ in three variables there exist polynomials $U$ and $V$ such that: $$W(x,y,z) = U(x,y,z)+V(x,y,z),$$ $$U(x,y,z) = U(y,x,z),$$ $$V(x,y,z) = -V(x,z,y).$$

Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.)

Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$. [i]Proposed by Navid Safaei [/i]

Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point $ j$ at which at least two disks are piled and then takes two disks from the point $ j$ and places one of them at $ j\minus{}1$ and one at $ j\plus{}1$. Initially, $ 2n\plus{}1$ disks are placed at point $ 0$. The player proceeds to perform moves as long as possible. Prove that after $ \frac{1}{6} n(n\plus{}1)(2n\plus{}1)$ moves no further moves will be possible and that at this stage, one disks remains at each of the positions $ \minus{}n,\minus{}n\plus{}1,...,0,...n$.

On the plane there are centrally symmetric convex polygon with area 1 and two his copies (each obtained from a polygon by some parallel transfer). It is known that no point of the plane is not covered by the three polygons at once. Prove that the total area covered by polygons, at least 2.

The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$ {A. Andjans, Riga)

A tape with width $ d < AB $ and edges perpendicular to $ AB $ moves in the plane of the acute-angled triangle $ ABC $. At what position of the tape will it cover the largest part of the triangle?

[b]p1.[/b] Compute $2017 + 7201 + 1720 + 172$. [b]p2. [/b]A number is called [i]downhill [/i]if its digits are distinct and in descending order. (For example, $653$ and $8762$ are downhill numbers, but $97721$ is not.) What is the smallest downhill number greater than 86432? [b]p3.[/b] Each vertex of a unit cube is sliced off by a planar cut passing through the midpoints of the three edges containing that vertex. What is the ratio of the number of edges to the number of faces of the resulting solid? [b]p4.[/b] In a square with side length $5$, the four points that divide each side into five equal segments are marked. Including the vertices, there are $20$ marked points in total on the boundary of the square. A pair of distinct points $A$ and $B$ are chosen randomly among the $20$ points. Compute the probability that $AB = 5$. [b]p5.[/b] A positive two-digit integer is one less than five times the sum of its digits. Find the sum of all possible such integers. [b]p6.[/b] Let $$f(x) = 5^{4^{3^{2^{x}}}}.$$ Determine the greatest possible value of $L$ such that $f(x) > L$ for all real numbers $x$. [b]p7.[/b] If $\overline{AAAA}+\overline{BB} = \overline{ABCD}$ for some distinct base-$10$ digits $A, B, C, D$ that are consecutive in some order, determine the value of $ABCD$. (The notation $\overline{ABCD}$ refers to the four-digit integer with thousands digit $A$, hundreds digit $B$, tens digit $C$, and units digit $D$.) [b]p8.[/b] A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube? [b]p9.[/b] Define $\lfloor x \rfloor$ as the greatest integer less than or equal to x, and ${x} = x - \lfloor x \rfloor$ as the fractional part of $x$. If $\lfloor x^2 \rfloor =2 \lfloor x \rfloor$ and $\{x^2\} =\frac12 \{x\}$, determine all possible values of $x$. [b]p10.[/b] Find the largest integer $N > 1$ such that it is impossible to divide an equilateral triangle of side length $ 1$ into $N$ smaller equilateral triangles (of possibly different sizes). [b]p11.[/b] Let $f$ and $g$ be two quadratic polynomials. Suppose that $f$ has zeroes $2$ and $7$, $g$ has zeroes $1$ and $ 8$, and $f - g$ has zeroes $4$ and $5$. What is the product of the zeroes of the polynomial $f + g$? [b]p12.[/b] In square $PQRS$, points $A, B, C, D, E$, and $F$ are chosen on segments $PQ$, $QR$, $PR$, $RS$, $SP$, and $PR$, respectively, such that $ABCDEF$ is a regular hexagon. Find the ratio of the area of $ABCDEF$ to the area of $PQRS$. [b]p13.[/b] For positive integers $m$ and $n$, define $f(m, n)$ to be the number of ways to distribute $m$ identical candies to $n$ distinct children so that the number of candies that any two children receive differ by at most $1$. Find the number of positive integers n satisfying the equation $f(2017, n) = f(7102, n)$. [b]p14.[/b] Suppose that real numbers $x$ and $y$ satisfy the equation $$x^4 + 2x^2y^2 + y^4 - 2x^2 + 32xy - 2y^2 + 49 = 0.$$ Find the maximum possible value of $\frac{y}{x}$. [b]p15.[/b] A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC, AC, AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the system of equations in integers: \[\begin{cases}z^x=y^{2x}\\ 2^z=4^x\\ x+y+z=20.\end{cases}\]

Determine all possible (finite or infinite) values of \[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\] if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying \[f(f(x))^4-f(f(x))+f(x)=1\] for all $x \in \mathbb{R}$.

[asy] /*Using regular asymptote, this diagram would take 30 min to make. Using cse5, this takes 5 minutes. Conclusion? CSE5 IS THE BEST PACKAGE EVER CREATED!!!!*/ size(100); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle); D(MP("$E$",EE,N)--MP("$D$",D,S)); D(D);D(EE); MA("80^\circ",8,D,EE,C,0.1); MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA("100^\circ",8,A,B,C,0.1,0); MA("60^\circ",8,C,A,B,0.1,0); //Credit to TheMaskedMagician for the diagram [/asy] In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals $\textbf{(A) }\frac{1}{4}\cos 10^\circ\qquad\textbf{(B) }\frac{\sqrt{3}}{8}\qquad\textbf{(C) }\frac{1}{4}\cos 40^\circ\qquad\textbf{(D) }\frac{1}{4}\cos 50^\circ\qquad\textbf{(E) }\frac{1}{8}$

Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point. [i]Vasile Pop[/i]

$\mathbb{R}^2$-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the $xy$ plane. The red player moves first. The first player to get $3$ of their points in a line without any of their opponent's points in between wins. What is the least number of moves in which Red can guarantee a win? (We count each time that Red places a point as a move, including when Red places its winning point.)

Let $x_1,x_2, ... , x_n$ be positive integers,Asumme that in their decimal representations no $x_i$ "prolongs" $x_j$.For instance , $123$ prolongs $12$ , $459$ prolongs $4$ , but $124$ does not prolog $123$. Prove that : $\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3$.

Let $a, b, c$ be positive reals. Prove that $$\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.$$

Let $ABCD$ be a convex quadrilateral. $O$ is the intersection of $AC$ and $BD$. $AO=3$ ,$BO=4$, $CO=5$, $DO=6$. $X$ and $Y$ are points in segment $AB$ and $CD$ respectively, such that $X,O,Y$ are collinear. The minimum of $\frac{XB}{XA}+\frac{YC}{YD}$ can be written as $\frac{a\sqrt{c}}{b}$ , where $\frac{a}{b}$ is in lowest term and $c$ is not divisible by any square number greater then $1$. What is the value of $10a+b+c$?

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.