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.

Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if [list] [*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and [*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and [*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list] Determine the number of good $p$-tuples.

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

p1. The parabola $y = ax^2 + bx$, $a < 0$, has a vertex $C$ and intersects the $x$-axis at different points $A$ and $B$. The line $y = ax$ intersects the parabola at different points $A$ and $D$. If the area of triangle $ABC$ is equal to $|ab|$ times the area of ​​triangle $ABD$, find the value of $ b$ in terms of $a$ without use the absolute value sign. p2. It is known that $a$ is a prime number and $k$ is a positive integer. If $\sqrt{k^2-ak}$ is a positive integer, find the value of $k$ in terms of $a$. p3. There are five distinct points, $T_1$, $T_2$, $T_3$, $T_4$, and $T$ on a circle $\Omega$. Let $t_{ij}$ be the distance from the point $T$ to the line $T_iT_j$ or its extension. Prove that $\frac{t_{ij}}{t_{jk}}=\frac{TT_i}{TT_k}$ and $\frac{t_{12}}{t_{24}}=\frac{t_{13}}{t_{34}}$ [img]https://cdn.artofproblemsolving.com/attachments/2/8/07fff0a36a80708d6f6ec6708f609d080b44a2.png[/img] p4. Given a $7$-digit positive integer sequence $a_1, a_2, a_3, ..., a_{2017}$ with $a_1 < a_2 < a_3 < ...<a_{2017}$. Each of these terms has constituent numbers in non-increasing order. Is known that $a_1 = 1000000$ and $a_{n+1}$ is the smallest possible number that is greater than $a_n$. As For example, we get $a_2 = 1100000$ and $a_3 = 1110000$. Determine $a_{2017}$. p5. At the oil refinery in the Duri area, pump-1 and pump-2 are available. Both pumps are used to fill the holding tank with volume $V$. The tank can be fully filled using pump-1 alone within four hours, or using pump-2 only in six hours. Initially both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $ b$ hours and continues again using only pump-2 for $c$ hours. If the operating cost of pump-1 is $15(a + b)$ thousand per hour and pump-2 operating cost is $4(a + c)$ thousand per hour, determine $ b$ and $c$ so that the operating costs of all pumps are minimum (express $b$ and $c$ in terms of $a$). Also determine the possible values ​​of $a$.

Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils and $216$ erasers were distributed. This distribution was made so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. In this case, how many notebooks, pencils and erasers did each family receive?

Lets say that a positive integer is $good$ if its equal to the the subtraction of two positive integer cubes. For example: $7$ is a $good$ prime because $2^3-1^3=7$. Determine how much the last digit of a $good$ prime may be worth. Give all the possibilities.

Determine the highest power of $2$ that divides $2^n!$.

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Prove that the set of all the points with both coordinates begin rational numbers can be written as a reunion of two disjoint sets $ A$ and $ B$ such that any line that that is parallel with $ Ox$, and respectively $ Oy$ intersects $ A$, and respectively $ B$ in a finite number of points.

Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.

Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.

Prove that the number $$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

Find all positive integers $m$ such that there exist positive integers $a_1,a_2,\ldots,a_{1378}$ such that: \[ m=\sum_{k=1}^{1378}{\frac{k}{a_k}}. \]

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Find all positive integers $n$ that can be represented in the form $$n=\mathrm{lcm}(a,b)+\mathrm{lcm}(b,c)+\mathrm{lcm}(c,a)$$ where $a,b,c$ are positive integers.

If $p$ is a prime positive integer and $x,y$ are positive integers, find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1) If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).

Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

[u]Round 1[/u] [b]p1.[/b] Jom and Terry both flip a fair coin. What is the probability both coins show the same side? [b]p2.[/b] Under the same standard air pressure, when measured in Fahrenheit, water boils at $212^o$ F and freezes at $32^o$ F. At thesame standard air pressure, when measured in Delisle, water boils at $0$ D and freezes at $150$ D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have $y = ax + b$. What is the value of $a + b$? (Ignore units.) [b]p3.[/b] What are the last two digits of $5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}$? [u]Round 2[/u] [b]p4.[/b] Compute the average of the magnitudes of the solutions to the equation $2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0$. [b]p5.[/b] How many integers between $1$ and $1000000$ inclusive are both squares and cubes? [b]p6.[/b] Simon has a deck of $48$ cards. There are $12$ cards of each of the following $4$ suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card? [u]Round 3[/u] [b]p7.[/b] Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability $p$ of shooting, and $1 - p$ of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, $p$ is constant. Once the ball has been shot, the game is over. What is the maximum value of $p$ that makes Christine’s total probability of shooting the ball $\frac{3}{20}$ ? [b]p8.[/b] If $x$ and $y$ are real numbers, then what is the minimum possible value of the expression $3x^2 - 12xy + 14y^2$ given that $x - y = 3$? [b]p9.[/b] Let $ABC$ be an equilateral triangle, let $D$ be the reflection of the incenter of triangle $ABC$ over segment $AB$, and let $E$ be the reflection of the incenter of triangle $ABD$ over segment $AD$. Suppose the circumcircle $\Omega$ of triangle $ADE$ intersects segment $AB$ again at $X$. If the length of $AB$ is $1$, find the length of $AX$. [u]Round 4[/u] [b]p10.[/b] Elaine has $c$ cats. If she divides $c$ by $5$, she has a remainder of $3$. If she divides $c$ by $7$, she has a remainder of $5$. If she divides $c$ by $9$, she has a remainder of $7$. What is the minimum value $c$ can be? [b]p11.[/b] Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a $10$-sided die (its faces are numbered from $1$ to $10$) $x$ times. If, within those $x$ rolls, the number $10$ appears, then you win. Assuming that you like winning, what is the highest value of $x$ where you would prefer to play the coin-flipping game over the die-rolling game? [b]p12.[/b] Let be the set $X = \{0, 1, 2, ..., 100\}$. A subset of $X$, called $N$, is defined as the set that contains every element $x$ of $X$ such that for any positive integer $n$, there exists a positive integer $k$ such that n can be expressed in the form $n = x^{a_1}+x^{a_2}+...+x^{a_k}$ , for some integers $a_1, a_2, ..., a_k$ that satisfy $0 \le a_1 \le a_2 \le ... \le a_k$. What is the sum of the elements in $N$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the smallest three-digit divisor of the number \[1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\]

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of integers. Define $a_n^{(0)} = a_n$ for all $n \in \mathbb{N}$. For all $M \geq 0$, we define $(a_n^{(M + 1)})_{n \in \mathbb{N}}:\, a_n^{(M + 1)} = a_{n + 1}^{(M)} - a_n^{(M)}, \forall n \in \mathbb{N}$. We say that $(a_n)_{n \in \mathbb{N}}$ is $\textrm{(M + 1)-self-referencing}$ if there exists $k_1$ and $k_2$ fixed positive integers such that $a_{n + k_1} = a_{n + k_2}^{(M + 1)}, \forall n \in \mathbb{N}$. (a) Does there exist a sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$? (a) Does there exist a stricly positive sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$?

Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.

Find all prime numbers that do not have a multiple ending in $2015$.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.