A sphere is the set of points at a fixed positive distance $r$ from its center. Let $\mathcal{S}$ be a set of $2010$-dimensional spheres. Suppose that the number of points lying on every element of $\mathcal{S}$ is a finite number $n$. Find the maximal possible value of $n$.

Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.

Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?

Give a prime number $p>2023$. Let $r(x)$ be the remainder of $x$ modulo $p$. Let $p_1<p_2< \ldots <p_m$ be all prime numbers less that $\sqrt[4]{\frac{1}{2}p}$. Let $q_1, q_2, \ldots, q_n$ be the inverses modulo $p$ of $p_1, p_2, \ldots p_n$. Prove that for every integers $0 < a,b < p$, the sets $$\{r(q_1), r(q_2), \ldots, r(q_m)\}, ~~ \{r(aq_1+b), r(aq_2+b), \ldots, r(aq_m+b)\}$$ have at most $3$ common elements.

Let $H$ be the orthocentre of an acute triangle $ABC$ with $BC > AC$, inscribed in a circle $\Gamma$. The circle with centre $C$ and radius $CB$ intersects $\Gamma$ at the point $D$, which is on the arc $AB$ not containing $C$. The circle with centre $C$ and radius $CA$ intersects the segment $CD$ at the point $K$. The line parallel to $BD$ through $K$, intersects $AB$ at point $L$. If $M$ is the midpoint of $AB$ and $N$ is the foot of the perpendicular from $H$ to $CL$, prove that the line $MN$ bisects the segment $CH$.

Let $A_1,A_2,...,A_n$ be $n$ finite subsets of a set $X, n \ge 2$, such that (i) $|A_i| \ge 2, 1 \le i \le n$, (ii) $ |A_i \cap A_j | \ne 1, j \le i < j \le n$. Prove that the elements of $A_1 \cup A_2 \cup ... \cup A_n$ may be colored with $2$ colors so that no $A_i$ is colored by the same color.

A $100 \times 100 \times 100$ cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces $100 \times 100$ of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a $\textit{column}$ a set of $100$ cubes forming a block $1 \times 1 \times 100$. Each of the $30 000$ columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly $k$ lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than $\frac {k} {100}$ switches on the red face.

There is a square of checkered paper measuring $102 \times 102$ squares and a connected figure of unknown shape, consisting of 101 cells. What is the largest number of such figures that can be cut from this square with a guarantee? A figure made up of cells is called [i]connected [/i] if any two its cells can be connected by a chain of its cells in which any two adjacent cells have a common side.

Let the operation * be defined by $ a * b \equal{} ab \plus{} a \minus{} b$. Which of the expression below is wrong, or are they all correct? A. $ a * a \equal{} a^2$ B. $ a * b \equal{} (\minus{}b) * (\minus{}a)$ C. $ (a * b) * a \equal{} a^2 * b$ D. $ (a * b) * c \equal{} (a * c) * b$ E. All four are correct

Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions: i) the number $n$ is composite; ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.

A die is an unitary cube with numbers from $1$ to $6$ written on its faces, so that each number appears once and the sum of the numbers on any two opposite faces is $7$. We construct a large $3 \cdot 3 \cdot 3$ cube using$ 27$ dice. Find all possible values of the sum of numbers which can be seen on the faces of the large cube.

Let $x$ satisfy $x^4+x^3+x^2+x+1=0$. Compute the value of $(5x+x^2)(5x^2+x^4)(5x^3+x^6)(5x^4+x^8)$. [i]Proposed by Andrew Zhao[/i]

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 $\textbf{(A) }24\qquad\textbf{(B) }33\qquad\textbf{(C) }45\qquad\textbf{(D) }54\qquad \textbf{(E) }57$