The integer n is a number expressed as the sum of an even number of different positive integers less than or equal to 2000. 1+2+ · · · +2000 Find all of the following positive integers that cannot be the value of n.

Consider the polynomials $P(x)=x^4-3x^3+x-3,\,\,\,\,Q(x)=x^2-2x-3 \,\,\,\, R(x)=-x^2-5x+a$ i) Find $a \in $R such that polynomial $R(x)$ is dividide by $x-2$ ii) Factor polynomials $P(x),Q(x)$ iii) Prove that exrpession $-x^2+x+\frac{P(x)}{Q(x)}+15$ is a perfect square.

This line graph represents the price of a trading card during the first $6$ months of $1993$. [asy] unitsize(18); for (int a = 0; a <= 6; ++a) { draw((4*a,0)--(4*a,10)); } for (int a = 0; a <= 5; ++a) { draw((0,2*a)--(24,2*a)); } draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5)); label("$Jan$",(2,0),S); label("$Feb$",(6,0),S); label("$Mar$",(10,0),S); label("$Apr$",(14,0),S); label("$May$",(18,0),S); label("$Jun$",(22,0),S); label("$\textbf{1993 PRICES FOR A TRADING CARD}$",(12,10),N); label("$\begin{tabular}{c}\textbf{P} \\ \textbf{R} \\ \textbf{I} \\ \textbf{C} \\ \textbf{E} \end{tabular}$",(-2,5),W); label("$1$",(0,2),W); label("$2$",(0,4),W); label("$3$",(0,6),W); label("$4$",(0,8),W); label("$5$",(0,10),W); [/asy] The greatest monthly drop in price occurred during $\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}$

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

Jane tells you that she is thinking of a three-digit number that is greater than $500$ that has exactly $20$ positive divisors. If Jane tells you the sum of the positive divisors of her number, you would not be able to figure out her number. If, instead, Jane had told you the sum of the \textit{prime} divisors of her number, then you also would not have been able to figure out her number. What is Jane's number? (Note: the sum of the prime divisors of $12$ is $2 + 3 = 5$, not $2 + 2 + 3 = 7$.)

Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

Prove that for positive real numbers $a, b, c$ the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*} When does equality occur?

How many positive integer solutions does the following system of equations have? $$\begin{cases}\sqrt{2020}(\sqrt{a}+\sqrt{b})=\sqrt{(c+2020)(d+2020)}\\\sqrt{2020}(\sqrt{b}+\sqrt{c})=\sqrt{(d+2020)(a+2020)}\\\sqrt{2020}(\sqrt{c}+\sqrt{d})=\sqrt{(a+2020)(b+2020)}\\\sqrt{2020}(\sqrt{d}+\sqrt{a})=\sqrt{(b+2020)(c+2020)}\\ \end{cases}$$

Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that 1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$, 2. $T_1 \subseteq T_2 \cup T_3$, 3. $T_2 \subseteq T_1 \cup T_3$, and 4. $T_3\subseteq T_1 \cup T_2$.

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$

For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) =8$. Prove that \[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\] When does equality hold? (Proposed by Stephan Wagner, Stellenbosch University)

Nine distinct light bulbs are placed in a circle, each of which is off. Determine the number of ways to turn on some of the light bulbs in the circle such that no four consecutive bulbs are all off.

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

Jesse has ten squares, which are labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?

For positive integers $i$ and $N$, let $k_{N,i}$ be the $i$th smallest positive integer such that the polynomial $\frac{x^2}{2023} + \frac{N_x}{7} - k_{N,i}$ has integer roots. Compute the minimum positive integer $N$ satisfying the condition $\frac{k_{N,2023}}{k_{N,1000}}< 3$. Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be $\max \left( 0, 25 \min \left( \frac{A}{E} , \frac{E}{A}\right)^{\frac32}\right)$, rounded to the nearest integer.

Is it possible to arrange natural numbers 1 to 8 on vertices of a cube such that each number divides sum of the three numbers sharing an edge with it?

Let $p$ be a prime. Prove that any complete graph with $1000p$ vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by $p$.

Let $a,b,c,d$ be the area of four faces of a tetrahedron, satisfying $a+b+c+d=1$. Show that $$\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2}$$ holds for all positive integers $n$.

A natural number $n$ is said to be [i]abundant [/i] if the sum of its divisors is greater than $2n$. For example, $18$ is abundant because the sum of its divisors, $1 + 2 + 3 + 6 + 9 + 18$, is greater than $36$. Write every even number greater than $46$ as a sum of two numbers abundant.

In the quadrilateral $ABCD$, $AD$ is parallel to $BC$. $K$ is a point on $AB$. Draw the line through $A$ parallel to $KC$ and the line through $B$ parallel to $KD$. Prove that these two lines intersect at some point on $CD$.

Let $ABCD$ be a trapezoid such that $AB \parallel CD, \angle{BAC}=25^\circ, \angle{ABC}=125^\circ,$ and $AB+AD=CD.$ Compute $\angle{ADC}.$

Given finitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains. (translated by L. Erdős)

In the $3\times5$ grid shown, fill in each empty box with a two-digit positive integer such that: [list][*]no number appears in more than one box, and [*] for each of the $9$ lines in the grid consisting of three boxes connected by line segments, the box in the middle of the line contains the least common multiple of the numbers in the two boxes on the line.[/list] 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.) [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.8) + fontsize(14); defaultpen(dps); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((6,0)--(7,0)--(7,1)--(6,1)--cycle); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((6,2)--(7,2)--(7,3)--(6,3)--cycle); draw((6,4)--(7,4)--(7,5)--(6,5)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((0.5,1)--(0.5,2)); draw((0.5,3)--(0.5,4)); draw((1,4)--(2,3)); draw((2.5,1)--(2.5,2)); draw((2.5,3)--(2.5,4)); draw((3,4)--(4,3)); draw((3,2)--(4,1)); draw((4.5,1)--(4.5,2)); draw((4.5,3)--(4.5,4)); draw((5,4.5)--(6,4.5)); draw((7,4.5)--(8,4.5)); draw((5,4)--(6,3)); draw((7,2)--(8,1)); draw((5,2)--(6,1)); draw((5,0.5)--(6,0.5)); draw((7,0.5)--(8,0.5)); draw((8.5,1)--(8.5,2)); draw((8.5,3)--(8.5,4)); label("$4$",(4.5, 0.5)); label("$9$",(8.5, 4.5)); [/asy]

Say that a positive integer is MPR (Math Prize Resolvable) if it can be represented as the sum of a 4-digit number MATH and a 5-digit number PRIZE. (Different letters correspond to different digits. The leading digits M and P can't be zero.) Say that a positive integer is MPRUUD (Math Prize Resolvable with Unique Units Digits) if it is MPR and the set of units digits $\{ \mathrm{H}, \mathrm{E} \}$ in the definition of MPR can be uniquely identified. Find the smallest positive integer that is MPR but not MPRUUD.