Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$. [i]Author: Zuming Feng and Oleg Golberg, USA[/i]

Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

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).