A cube $A_1A_2A_3A_4A_5A_6A_7A_8$ is given in space. We will mark its center with the letter $S$ (intersection of solid diagonals). Find all natural numbers $k$ for which there exists a plane not containing the point $S$ and intersecting just $k$ of the rays $SA_1, SA_2, .. SA_8$

A student of the National Technical University was reading advanced mathematics last summer for 37 days according to the following rules : (a) He was reading at least one hour every day. (b) He was reading an integer number of hours, but not more than 12, each day. (c) He had to read at most 60 hours in total. Prove that there were some successive days during which the student was reading exactly 13 hours in total.

Call a positive integer [i]descending[/i] if, reading left to right, each of its digits (other than its leftmost) is less than or equal to the previous digit. For example, $4221$ and $751$ are descending while $476$ and $455$ are not descending. Determine whether there exists a positive integer $n$ for which $16^n$ is descending.

Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that : (i) $|A| = 12$ (ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.

Let $S_n$ be the area of the figure enclosed by a curve $y=x^2(1-x)^n\ (0\leq x\leq 1)$　and the $x$-axis. Find $\lim_{n\to\infty} \sum_{k=1}^n S_k.$

You roll three six-sided dice. If the three dice and indistinguishable, how many combinations of numbers can result?

Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

[b]p1.[/b] Compute $45\times 45 - 6$. [b]p2.[/b] Consecutive integers have nice properties. For example, $3$, $4$, $5$ are three consecutive integers, and $8$, $9$, $10$ are three consecutive integers also. If the sum of three consecutive integers is $24$, what is the smallest of the three numbers? [b]p3.[/b] How many positive integers less than $25$ are either multiples of $2$ or multiples of $3$? [b]p4.[/b] Charlotte has $5$ positive integers. Charlotte tells you that the mean, median, and unique mode of his five numbers are all equal to $10$. What is the largest possible value of the one of Charlotte's numbers? [b]p5.[/b] Mr. Meeseeks starts with a single coin. Every day, Mr. Meeseeks goes to a magical coin converter where he can either exchange $1$ coin for $5$ coins or exchange $5$ coins for $3$ coins. What is the least number of days Mr. Meeseeks needs to end with $15$ coins? [b]p6.[/b] Twelve years ago, Violet's age was twice her sister Holo's age. In $7$ years, Holo's age will be $13$ more than a third of Violet's age. $3$ years ago, Violet and Holo's cousin Rindo's age was the sum of their ages. How old is Rindo? [b]p7.[/b] In a $2 \times 3$ rectangle composed of $6$ unit squares, let $S$ be the set of all points $P$ in the rectangle such that a unit circle centered at $P$ covers some point in exactly $3$ of the unit squares. Find the area of the region $S$. For example, the diagram below shows a valid unit circle in a $2 \times 3$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/d/9/b6e00306886249898c2bdb13f5206ced37d345.png[/img] [b]p8.[/b] What are the last four digits of $2^{1000}$? [b]p9.[/b] There is a point $X$ in the center of a $2 \times 2 \times 2$ box. Find the volume of the region of points that are closer to $X$ than to any of the vertices of the box. [b]p10.[/b] Evaluate $\sqrt{37 \cdot 41 \cdot 113 \cdot 290 - 4319^2}$ [b]p11.[/b] (Estimation) A number is abundant if the sum of all its divisors is greater than twice the number. One such number is $12$, because $1+2+3+4+6+12 = 28 > 24$: How many abundant positive integers less than $20190$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

Calculate the sum $\frac{5}{2.7}+\frac{5}{7.12}+...+\frac{5}{2002.2007}$

There are $2024$ cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities $A, B, C, X, Y, Z$, it is possible to fly directly from some of the cities $A, B, C$ to some of the cities $X, Y, Z$. Prove that it is possible to plan a route $T_1\to T_2 \to \ldots \to T_{2022}$ that passes through $2022$ distinct cities. [i]Proposed by Lior Shayn[/i]

The number of scalene triangles having all sides of integral lengths, and perimeter less than $ 13$ is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$

Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $ [i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]

Let $A \in \mathcal{M}_n(\mathbb{C})$ be a matrix such that $(AA^*)^2=A^*A$, where $A^*=(\bar{A})^t$ denotes the Hermitian transpose (i.e., the conjugate transpose) of $A$. (a) Prove that $AA^*=A^*A$. (b) Show that the non-zero eigenvalues of $A$ have modulus one.

If $b>1$, $\sin x>0$, $\cos x>0$, and $\log_b \sin x = a$, then $\log_b \cos x$ equals $\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}$

Find all pairs of real numbers $(a, b)$ such that the equality $$|ax+by|+ |bx + ay| = 2|x| + 2|y|$$ holds for all reals $x$ and $y$.

In $ \triangle ABC, \ \cos(2A \minus{} B) \plus{} \sin(A\plus{}B) \equal{} 2$ and $ AB\equal{}4.$ What is $ BC?$ $ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$

The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.

Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$

In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90$° $ -2 \alpha $. [i]Proposed by Dusan Djukic[/i]

Prove that all continuous solutions of the functional equation $\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)$ are the constant functions.

The natural number $n>1$ is such that there exist $a\in \mathbb{N}$ and a prime number $q$ which satisfy the following conditions: 1) $q$ divides $n-1$ and $q>\sqrt{n}-1$ 2) $n$ divides $a^{n-1}-1$ 3) $gcd(a^\frac{n-1}{q}-1,n)=1$. Is it possible for $n$ to be a composite number?

A $6$ meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?

On a certain board, fractions are always written in their lowest form. Pionaj starts with 2 random positive fractions. After every minute,he replaces one of the previous 2 fractions (at random) with a new fraction that is equal to the sum of their numerators divided by the sum of their denominators. Given that he continues this indefinitely, show that eventually all the resulting fractions would be in their lowest forms even before writing them on the board(recall that he has to reduce each fration to their lowest form beore writing it on the board for the next operation). (for example starting with $\frac{15}{7}$ and $\frac{10}{3}$ he may replace it with $\frac{5}{2}$