Placing $n \in {\mathbb N}$ circles with radius $1$ $unit$ inside a square with side $100$ $unit$ such that, whichever line segment with lenght $10$ $unit$ intersect at least one circle. Prove that $$n \geq 416$$

We work in the decimal system and the following operations are allowed to be done with a positive integer: a) append $4$ at the end of the number. b) append $0$ at the end of the number. c) divide the number by $2$ if it's even. Show that starting with $4$, we can reach every positive integer by a finite number of these operations

Define $a_n$: the coefficient of then item $x$ in $(3-\sqrt{x})^n$, where $n$ is a positive integer. Then $\lim_{n\to\infty}\left(\frac{3^2}{a_2}+\frac{3^3}{a_3}+\cdots+\frac{3^n}{a_n}\right)=$________.

Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?

A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between them. Determine all the values of $ n$ not exceeding 10 for which this arrangement is possible. For which values of $ n$ is it impossible?

How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.)

Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that a) For every integer $n$, we have $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$ b) The value $\frac{1}{2\sqrt{17} }$ is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$

As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction $\frac{19}{95}$. After crossing out $9$ in the numerator and denominator (“reduction” by $9$), we get $\frac{1}{5}$ which is the correct answer. In the same way, a fraction $\frac{1999}{9995}$ can be “reduced” by three nines (cross out $999$ in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to $\frac13$ ? (We consider fractions of the form $\frac{1a}{a3}$. Here, with the letter $a$ we denote several numbers that follow in the same order in the numerator after $1$, and in the denominator before $3$. “Reduce” by $a$.)

Let $\omega$ be a primitive $7$th root of unity. Find $$\prod_{k=0}^6\left(1+\omega^k-\omega^{2k}\right).$$ (A complex number is a primitive root of unity if and only if it can be written in the form $e^{2k\pi i/n}$, where $k$ is relatively prime to $n$.)

Let $\triangle ABC$ with $AB=AC$ and $BC=14$ be inscribed in a circle $\omega$. Let $D$ be the point on ray $BC$ such that $CD=6$. Let the intersection of $AD$ and $\omega$ be $E$. Given that $AE=7$, find $AC^2$. [i]Proposed by Ephram Chun and Euhan Kim[/i]

Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that: $ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$

Let $ABC$ be an acute triangle with circumcenter $O$. Let $A'$ be the center of the circle passing through $C$ and tangent to $AB$ at $A$, let $B'$ be the center of the circle passing through $A$ and tangent to $BC$ at $B$, let $C'$ be the center of the circle passing through $B$ and tangent to $CA$ at $C$. a) Prove that the area of triangle $A'B'C'$ is not less than the area of triangle $ABC$. b) Let $X, Y, Z$ be the projections of $O$ onto lines $A'B', B'C', C'A'$. Given that the circumcircle of triangle $XYZ$ intersects lines $A'B', B'C', C'A'$ again at $X', Y', Z'$ ($X' \neq X, Y' \neq Y, Z' \neq Z$), prove that lines $AX', BY', CZ'$ are concurrent.

The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.

In the net drawn below, in how many ways can one reach the point $3n+1$ starting from the point $1$ so that the labels of the points on the way increase? [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.66,ymax=6.3; draw((1,2)--(xmax,0*xmax+2)); draw((1,0)--(xmax,0*xmax+0)); draw((0,1)--(1,2)); draw((1,0)--(0,1)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((5,2)--(7,0)); draw((5,0)--(7,2)); draw((7,2)--(9,0)); draw((7,0)--(9,2)); dot((1,0),linewidth(1pt)+ds); label("2",(0.96,-0.5),NE*lsf); dot((0,1),linewidth(1pt)+ds); label("1",(-0.42,0.9),NE*lsf); dot((1,2),linewidth(1pt)+ds); label("3",(0.98,2.2),NE*lsf); dot((2,1),linewidth(1pt)+ds); label("4",(1.92,1.32),NE*lsf); dot((3,2),linewidth(1pt)+ds); label("6",(2.94,2.2),NE*lsf); dot((4,1),linewidth(1pt)+ds); label("7",(3.94,1.32),NE*lsf); dot((6,1),linewidth(1pt)+ds); label("10",(5.84,1.32),NE*lsf); dot((3,0),linewidth(1pt)+ds); label("5",(2.98,-0.46),NE*lsf); dot((5,2),linewidth(1pt)+ds); label("9",(4.92,2.24),NE*lsf); dot((5,0),linewidth(1pt)+ds); label("8",(4.94,-0.42),NE*lsf); dot((8,1),linewidth(1pt)+ds); label("13",(7.88,1.34),NE*lsf); dot((7,2),linewidth(1pt)+ds); label("12",(6.8,2.26),NE*lsf); dot((7,0),linewidth(1pt)+ds); label("11",(6.88,-0.38),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Let $b = \tfrac 12 (-1 + 3\sqrt{5})$. Determine the number of rational numbers which can be written in the form \[ a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a_1b + a_0 \] where $a_0, a_1, \dots, a_{2014}$ are nonnegative integers less than $b$. [i]Proposed by Michael Kural and Evan Chen[/i]

If, in a triangle of sides $a, b, c$, the incircle has radius $\frac{b+c-a}{2}$, what is the magnitude of $\angle A$?

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive. [i]Proposed by Andy Xu[/i]

Let $I$ be the incenter of triangle $ABC$; $H_B, H_C$ the orthocenters of triangles $ACI$ and $ABI$ respectively; $K$ the touching point of the incircle with the side $BC$. Prove that $H_B, H_C$ and K are collinear. [i]Proposed by M.Plotnikov[/i]

The grid to the right consists of 74 unit squares, marked by gridlines. Partition the grid into five regions along the gridlines so that the areas of the regions are 1, 13, 19, 20, and 21. A square with a number should be contained in the region with that area. [asy]unitsize(20); path p = (5,0)--(3,0)--(3,1)--(1,1)--(1,2)--(0,2)--(0,7)--(1,7)--(1,8)--(3,8)--(3,9)--(5,9); draw(p^^reflect((5,0),(5,3.14))*p); int[] v = {0,1,1,0,0,0,0,0,1,1}; int[] h = {0,1,0,0,0,0,0,1,1}; for(int i=1; i<10; ++i) { draw((i,v[i])--(i,9-v[i]),dotted); } for(int i=1; i<9; ++i) { draw((h[i],i)--(10-h[i],i),dotted); } int[][] dord = {{1,4,21},{2,4,19},{3,4,1},{4,4,13},{5,4,20},{6,4,19},{7,4,20},{8,4,19},{2,2,21},{2,6,19},{4,7,19},{5,1,21},{7,6,13},{7,2,13}}; for(int i=0; i<14; ++i){ label(string(dord[i][2]),(dord[i][0]+.5,.5+dord[i][1])); } [/asy] You do not need to prove that your answer is the only one possible; you merely need to find an answer 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.)

Let $ABC$ be an acute, non isosceles triangle with the orthocenter $H$, circumcenter $O$ and $AD$ is the diameter of $(O)$. Suppose that the circle $(AHD)$ meets the lines $AB, AC$ at $F$, respectively. Denote $J, K$ as orthocenter and nine- point center of $AEF$. Prove that $HJ \parallel BC$ and $KO = KH$.

Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$

$n\geq3$ boxes are placed around a circle. At the first step we choose some boxes. At the second step for each chosen box we put a ball into the chosen box and into each of its two neighbouring boxes. Find the total number of possible distinct ball distributions which can be obtained in this way. (All balls are identical.)