Found problems: 155
What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.
Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$.
Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$.
Determine the largest possible value of $T$.
The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$.
Denote by $d_i$ the distance from a point $P$ to $\ell_i$.
For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?
Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.
Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$.
Determine the maximum value of $M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$
Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$.
Example:
[img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]
What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$?
A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$
Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$.
Determine the minimum and the maximum of $x$ in terms of $n$.
Let $a(x)$ and $b(x)$ be continuous functions on $[0,1]$ and let $0 \leq a(x) \leq a <1$ on that range. Under what other conditions (if any) is the solution of the equation for $u,$
$$ u= \max_{0 \leq x \leq 1} b(x) +a(x)u$$
given by
$$u = \max_{0 \leq x \leq 1} \frac{b(x)}{1-a(x)}.$$
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$.
($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).
The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$.
a) Determine the smallest and the largest possible values of $xy + yz - xz$.
b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).
In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour.
What is the largest possible value of $n$?
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?
The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?
Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is
(A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$.
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?
[i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.