A number is called [i]good[/i] if it can be written as the sum of the squares of three consecutive positive integers. A number is called excellent if it can be written as the sum of the squares of four consecutive positive integers. (For instance, $14 = 1^2 + 2^2 + 3^2$ is good and $30 =1^2 +2^2 +3^2+4^2$ is excellent.) A good number $G$ is called [i]splendid[/i] if there exists an excellent number $E$ such that $3G-E = 2025.$ If the sum of all splendid numbers is $S,$ find the remainder when $S$ is divided by $1000.$

Let $f$ of degree at most 13 such that $f(k) = 13^k$ for $0 \leq k \leq 13$. Compute the last three digits of $f(14)$. [i]Proposed by Kaylee Ji[/i]

Determine all pairs of real numbers $(k, d)$ such that the system of equations $$\begin{cases} x^3 + y^3 = 2 \\ kx + d = y\end{cases}$$ has no solutions $(x, y)$ with $x$ and $y$ real numbers.

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

Let positive numbers $a_1, a_2, ..., a_{3n}$ $(n \geq 2)$ constitute an arithmetic progression with common difference $d > 0$. Prove that among any $n + 2$ terms in this progression, there exist two terms $a_i, a_j$ $(i \neq j)$ satisfying $1 < \frac{|a_i - a_j|}{nd} < 2$.

Pasha placed numbers from 1 to 100 in the cells of the square $10\times 10$, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green?

Let $a, b, c, d, e, f, g, h, i$ be distinct integers from $1$ to $9$. The minimum possible positive value of $$\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}$$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that \[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}). \] [i]Proposed by Romania[/i]

Let $ABC$ be a scalene triangle. Let $\ell$ be a line passing through point $B$ that lies outside of the triangle $ABC$ and creates different angles with sides $AB$ and $BC$ . The point $M$ is the midpoint of side $AC$ and the ponts $H_a$ and $H_c$ are the bases of the perpendicular lines on the line $\ell$ drawn from points $A$ and $C$ respectively. The circle circumscribing triangle $MBH_a$ intersects AB at the point $A_1$ and the circumscribed circle of triangle $MBH_c$ intersects $BC$ at point $C_1$. The point $A_2$ is symmetric to the point $A$ relative to the point $A_1$ and the point $C_2$ is symmetric to the point $C_1$ relative to the point $C_1$. Prove that the lines $\ell$, $AC_2$ and $CA_2$ intersect at one point. (Yana Kolodach)

Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? $ \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $

Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. $5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?

In the lower-left $3\times3$ square of an $8\times8$ chessboard there are nine pawns. Every pawn can jump horizontally or vertically over a neighboring pawn to the cell across it if that cell is free. Is it possible to arrange the nine pawns in the upperleft $3\times3$ square of the chessboard using finitely many such moves?

We define the following sequences: • Sequence $A$ has $a_n = n$. • Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise. • Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$ .• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise. • Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$ Prove that the terms of sequence E are exactly the perfect cubes.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ such that $AB=BC=CD$. Let $M$ and $N$ be the midpoints of $AD$ and $AB$ respectively. The line $CM$ meets $\Gamma$ again at $E$. Prove that the tangent at $E$ to $\Gamma$, the line $AD$ and the line $CN$ are concurrent.

Let $P$ be a point on the median $AM$ of a triangle $ABC$. Suppose that the tangents to the circumcircles of $ABP$ and $ACP$ at $B$ and $C$ respectively meet at $Q$. Show that $\angle PAB = \angle CAQ$.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Points $ A,B,C$ and $ D$ lie on a line, in that order, with $ AB\equal{}CD$ and $ BC\equal{}12$. Point $ E$ is not on the line, and $ BE\equal{}CE\equal{}10$. The perimeter of $ \triangle AED$ is twice the perimeter of $ \triangle BEC$. Find $ AB$. $ \text{(A)}\ 15/2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 17/2 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 19/2$

A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.

Fill in each empty cell of the grid with a digit from 1 to 8 so that every row and every column contains each of these digits exactly once. Some diagonally adjacent cells have been joined together. For these pairs of joined cells, the same number must be written in both. [asy] filldraw((0,0)--(0,8)--(8,8)--(8,0)--cycle,white); path removex(pair p) { return ((p.x-0.5, p.y)--(p.x+0.5,p.y)); } path removey(pair p) { return ((p.x, p.y-0.5)--(p.x,p.y+0.5)); } unitsize(1cm); draw((0,0)--(8,0)--(8,8)--(0,8)--cycle, linewidth(2)); for(int i = 0; i < 8; ++i){ draw((0,i)--(8,i)); } for(int j = 0 ; j<8; ++j){ draw((j,0)--(j,8)); } pair [] pointsa = {(1,2),(3,1),(5,7),(7,6)}; pair [] pointsb= {(1,5),(4,4),(2,7),(6,1),(7,3)}; for(int q = 0; q<4; ++q){ draw(removex(pointsa[q]), white+linewidth(2)); draw(removey(pointsa[q]),white+linewidth(2)); draw(arc(pointsa[q]+(0.5,-0.5),0.5,90,180)); draw(arc(pointsa[q]-(0.5,-0.5),0.5,270,0,CCW)); draw(pointsa[q]+(-0.5,0)--pointsa[q]+(-1,0)); draw(pointsa[q]+(0.5,0)--pointsa[q]+(1,0)); draw(pointsa[q]+(0,0.5)--pointsa[q]+(0,1)); draw(pointsa[q]+(0,-0.5)--pointsa[q]+(0,-1)); } for(int q = 0; q<5; ++q){ draw(removex(pointsb[q]), white+linewidth(2)); draw(removey(pointsb[q]),white+linewidth(2)); draw(arc(pointsb[q]+(0.5,0.5),0.5,180,270,CCW)); draw(arc(pointsb[q]-(0.5,0.5),0.5,0,90,CCW)); draw(pointsb[q]+(-0.5,0)--pointsb[q]+(-1,0)); draw(pointsb[q]+(0.5,0)--pointsb[q]+(1,0)); draw(pointsb[q]+(0,0.5)--pointsb[q]+(0,1)); draw(pointsb[q]+(0,-0.5)--pointsb[q]+(0,-1)); } int [][] x = { {1,0,0,0,0,0,0,0}, {2,3,0,0,0,0,0,0}, {0,4,5,0,0,0,0,0}, {0,0,6,0,1,0,0,0}, {0,0,0,7,0,1,0,0}, {0,0,0,0,0,3,4,0}, {0,0,0,0,0,0,2,8}, {0,0,0,0,0,0,0,5} }; for(int k = 0; k<8; ++k){ for(int l = 0; l<8; ++l){ if(x[k][l]!=0){ label(string(x[k][l]), (l+0.5,-k+7.5), fontsize(24pt)); } } } [/asy] There is a unique solution, but 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.)

A polynomial $P (x)$ with real coefficients and of degree $n \ge 3$ has $n$ real roots $x_1 <x_2 < \cdots < x_n$ such that \[x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} \] Prove that the maximum value of $|P (x)|$ on the interval $[x_1 , x_n ]$ is attained in the interval $[x_{n-1} , x_n ]$.

Each BMT, every student chooses one of three focus rounds to take. Bob plans to attend BMT for the next $4$ years and wants to gure out what focus round to take each year. Given that he wants to take each focus round at least once, how many ways can he choose which round to take each year?

Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.