In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides. Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle. [img]https://1.bp.blogspot.com/-KWrhwMHXfDk/XzcIkhWnK5I/AAAAAAAAMYk/Tj6-PnvTy9ksHgke8cDlAjsj2u421Xa9QCLcBGAsYHQ/s0/1993%2BMohr%2Bp4.png[/img]

Find all integers $n$, such that the following number is a perfect square \[N= n^4 + 6n^3 + 11n^2 +3n+31. \]

For real numbers $a$ and $b$, define the sequence $\{x_{a,b}(n)\}$ as follows: $x_{a,b}(1)=a$, $x_{a,b}(2)=b$, and for $n>1$, $x_{a,b}(n+1)=(x_{a+b}(n-1))^2+(x_{a,b}(n))^2$. For real numbers $c$ and $d$, define the sequence $\{y_{c,d}(n)\}$ as follows: $y_{c,d}(1)=c$, $y_{c,d}(2)=d$, and for $n>1$, $y_{c,d}(n+1)=(y_{c,d}(n-1)+y_{c,d}(n))^2$. Call $(a,b,c)$ a good triple if there exists $d$ such that for all $n$ sufficiently large, $y_{c,d}(n)=(x_{a,b}(n))^2$. For some $(a,b)$ there are exactly three values of $c$ that make $(a,b,c)$ a good triple. Among these pairs $(a,b)$, compute the maximum value of $\lfloor 100(a+b)\rfloor$.

A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$: $A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$. However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.

[b]1.[/b] Let $C_{ij}$ ($i,j=1,2,3$) be the coefficients of a real non-involutive orthogonal transformation. Prove that the function $w= \sum_{i,j=1}^{3} c_{ ij}z_{i}\bar{z_{ j}}$ maps the surface of complex unit sphere $\sum_{i=1}^{3} z_{i}\bar{z_{i}} = 1$ onto a triangle of the w-plane. [b](F. 3)[/b]

How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$.

In the $8\times 8$ grid shown, fill in $12$ of the grid cells with the numbers $1-12$ so that the following conditions are satisfied: [list] [*]Each cell contains at most one number, and each number from $1-12$ is used exactly once. [*]Two cells that both contain numbers may not touch, even at a point. [*]A clue outside the grid pointing at a row or column gives the sum of all the numbers in that row or column. Rows and columns without clues have an unknown sum.[/list] You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(150); defaultpen(linewidth(0.8)); path arrow=(-1/8,1/8)--(1/8,0)--(-1/8,-1/8)--cycle; int sumRows[]={3,13,20,0,21,0,18,3}; int sumCols[]={24,1,3,0,20,13,0,11}; for(int i=0;i<=8;i=i+1) draw((i,0)--(i,8)^^(0,i)--(8,i)); for(int j=0;j<=7;j=j+1) { if(sumRows[j]>0) { filldraw(shift(-1/4,j+1/2)*arrow,black); label("$"+(string)sumRows[j]+"$",(-7/8,j+1/2)); } if(sumCols[j]>0) { filldraw(shift(j+1/2,8+3/8)*(rotate(270,origin)*arrow),black); label("$"+(string)sumCols[j]+"$",(j+1/2,9)); } } [/asy]

Prove that if $ k $ is a natural number, then $$ (1 + x)(1 + x^2) (1 + x^4) \ldots (1 + x^{2^k}) =1 + x + x^2 + x^3+ \ldots + x^m$$ where $ m $ is a natural number dependent on $ k $; determine $ m $.

Which of the following is the largest? $\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{3}{8} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{7}{24}$

Let $ABC$ be an acute triangle with $\angle BAC>\angle BCA$, and let $D$ be a point on side $AC$ such that $|AB|=|BD|$. Furthermore, let $F$ be a point on the circumcircle of triangle $ABC$ such that line $FD$ is perpendicular to side $BC$ and points $F,B$ lie on different sides of line $AC$. Prove that line $FB$ is perpendicular to side $AC$ .

Find all pairs of coprime natural numbers $a$ and $b$ such that the fraction $\frac{a}{b}$ is written in the decimal system as $b.a.$

Let $p$ be a prime number greater than $5$. Prove that there exists a positive integer $n$ such that $p$ divides $20^n+ 15^n-12^n$.

Let $PQRS$ be a quadrilateral that has an incircle and $PQ\neq QR$. Its incircle touches sides $PQ,QR,RS,$ and $SP$ at $A,B,C,$ and $D$, respectively. Line $RP$ intersects lines $BA$ and $BC$ at $T$ and $M$, respectively. Place point $N$ on line $TB$ such that $NM$ bisects $\angle TMB$. Lines $CN$ and $TM$ intersect at $K$, and lines $BK$ and $CD$ intersect at $H$. Prove that $\angle NMH=90^{\circ}$.

A non-self intersecting hexagon $RANDOM$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points $$(0,0), (10,0), (10,10), (0,10), (3,4), (6,2).$$ Let $a_{\text{max}}$ be the greatest possible area of $RANDOM$ and $a_{\text{min}}$ the least possible area of $RANDOM.$ Find $a_{\text{max}}-a_{\text{min}}.$

Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$

The $2011$th prime number is $17483$ and the next prime is $17489$. Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

Let $ABC$ be an acute-angled scalene triangle and $T$ be a point inside it such that $\angle ATB = \angle BTC = 120^o$. A circle centered at point $E$ passes through the midpoints of the sides of $ABC$. For $B, T, E$ collinear, find angle $ABC$.

A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being $30^o$. A ball is played from just in front of the pocket at the $30^o$. vertex toward the midpoint of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the $60^o$ vertex after $8$ reflections.

The fraction $\frac{3}{10}$ can be written as the sum of two positive fractions with numerator $1$ as follows: $\frac{3}{10} =\frac{1}{5}+\frac{1}{10}$ and also $\frac{3}{10}=\frac{1}{4}+\frac{1}{20}$. There are the only two ways in which this can be done. In how many ways can $\frac{3}{1984}$ be written as the sum of two positive fractions with numerator $1$? Is there a positive integer $n,$ not divisible by $3$, such that $\frac{3}{n}$ can be written as the sum of two positive fractions with numerator $1$ in exactly $1984$ ways?

Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus.

Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that $$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$

A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.

Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, . Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?