Let $p$ be a prime. We say $p$ is [i]good[/i] if and only if for any positive integer $a,b,$ such that $$a\equiv b (\textup{mod}p)\Leftrightarrow a^3\equiv b^3 (\textup{mod}p).$$Prove that (1)There are infinite primes $p$ which are [i]good[/i]; (2)There are infinite primes $p$ which are not [i]good[/i].

Find the solutions in prime numbers of the following equation $p^4 + q^4 + r^4 + 119 = s^2 .$

Express the sum $S_m(n)=1^m+2^m+\ldots +(n-1)^m$ with Bernolli numbers.

Find all pairs of two positive integers $(m,n)$ satisfying $ 3^m - 7^n = 2 $.

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

Among $18$ parts placed in a row, some three in a row weigh $99 $ g each, and all the rest weigh $100$ g each. On a scale with an arrow, identify all $99$-gram parts.

Ruby writes the numbers $1, 2, 3, . . . , 10$ on the whiteboard. In each move, she selects two distinct numbers, $a$ and $b$, erases them, and replaces them with $a+b-1$. She repeats this process until only one number, $x$, remains. What are all the possible values of $x$?

Consider all functions $f:\mathbb{N}\to\mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.

The two tangents to the incircle of a right-angled triangle $ABC$ that are perpendicular to the hypotenuse $AB$ intersect it at the points $P$ and $Q$. Find $\angle PCQ$. (M Evdokimov,)

Circles $\text{A}$ and $\text{B}$ are concentric, with the radius of $\text{A}$ being $\sqrt{17}$ times the radius of $B$. The largest line segment that can be draw in the region bounded by the two circles has length $32$. Compute the radius of circle $B$. [center][img]https://cdn.artofproblemsolving.com/attachments/7/4/6bc4aed9842cdfbeb95853d508a22b61a10c9c.png[/img][/center]

Chords $AC$ and $BD$ of a circle $w$ intersect at $E$. A circle that is internally tangent to $w$ at a point $F$ also touches the segments $DE$ and $EC$. Prove that the bisector of $\angle AFB$ passes through the incenter of $\triangle AEB$.

A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least $55$. The minimum number of red chips is $\textbf{(A) }24\qquad\textbf{(B) }33\qquad\textbf{(C) }45\qquad\textbf{(D) }54\qquad \textbf{(E) }57$

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

$A_1A_2A_3A_4A_5A_6A_7A_8$ is a regular octagon. Let $P$ be a point inside the octagon such that the distances from $P$ to $A_1A_2, A_2A_3$ and $A_3A_4$ are $24, 26$ and $27$ respectively. The length of $A_1A_2$ can be written as $a \sqrt{b} -c$, where $a,b$ and $c$ are positive integers and $b$ is not divisible by any square number other than $1$. What is the value of $(a+b+c)$?

Determine every prime numbers $p$ and $q , p \le q$ for which $pq | (5^p - 2^ p )(7^q -2 ^q )$

Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.

An escalator (moving staircase) of $ n$ uniform steps visible at all times descends at constant speed. Two boys, $ A$ and $ Z$, walk down the escalator steadily as it moves, $ A$ negotiating twice as many escalator steps per minute as $ Z$. $ A$ reaches the bottom after taking $ 27$ steps while $ Z$ reaches the bottom after taking $ 18$ steps. Then $ n$ is: $ \textbf{(A)}\ 63 \qquad \textbf{(B)}\ 54 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 30$

Jane has two bags $X$ and $Y$. Bag $X$ contains 4 red marbles and 5 blue marbles (and nothing else). Bag $Y$ contains 7 red marbles and 6 blue marbles (and nothing else). Jane will choose one of her bags at random (each bag being equally likely). From her chosen bag, she will then select one of the marbles at random (each marble in that bag being equally likely). What is the probability that she will select a red marble?

An equilateral triangle $ A_1 A_2 A_3$ is divided into four smaller equilateral triangles by joining the midpoints $ A_4,A_5,A_6$ of its sides. Let $ A_7,...,A_{15}$ be the midpoints of the sides of these smaller triangles. The $ 15$ points $ A_1,...,A_{15}$ are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points $ A_i,A_j,A_k$ having the same color.

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city? $ \textbf{(A)}\ 22\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$

The $U$-tile is made up of $1 \times 1$ squares and has the following shape: [img]https://cdn.artofproblemsolving.com/attachments/8/7/5795ee33444055794119a99e675ef977add483.png[/img] where there are two vertical rows of a squares, one horizontal row of $b$ squares, and also $a \ge 2$ and $b \ge 3$. Notice that there are different types of tile $U$ . For example, some types of $U$ tiles are as follows: [img]https://cdn.artofproblemsolving.com/attachments/0/3/ca340686403739ffbbbb578d73af76e81a630e.png[/img] Prove that for each integer $n \ge 6$, the board of $n\times n$ can be completely covered with $U$-tiles , with no gaps and no overlapping clicks. Clarifications: The $U$-tiles can be rotated. Any amount can be used in the covering of tiles of each type.

Choose arbitrarily $n$ vertices of a regular $2n-$gon and colour them red. The remaining vertices are coloured blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.

If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: $ \textbf{(A) }10\qquad\textbf{(B) }9 \qquad\textbf{(C) }2\qquad\textbf{(D) }-2\qquad\textbf{(E) }-9 $

For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.

Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.