The leg $a$ and the hypotenuse of a right triangle are in the ratio $12:13$. If we shorten the hypotenuse by $23~\mathrm{cm}$ and the leg $a$ by $27~\mathrm{cm}$, we get a new right triangle, the length of the second leg of which has not changed. Calculate the sides of the original triangle.
The diameter of the front wheel is $1,1~\mathrm{m}$, and the rear wheel is $0,8~\mathrm{m}$. What is the distance traveled if the first wheel made 69 revolutions less than the last? Round the result to the nearest centimeter. Use approximation $\frac{22}{7}$ for $\pi$. Write down the answer.
Let $n \ge 2$ be an integer and $(a_1, a_2, \dots, a_n)$ be a permutation of $1, 2, \dots, n$. For each $1 \le k \le n$, $a_k$ apples are placed at the point $k$ on the real axis. Children named $A, B, C$ are assigned points $x_A, x_B, x_C \in \{1, 2, \dots, n\}$, respectively. For each $k \in \{1, 2, \dots, n\}$, children whose points are closest to $k$ divide $a_k$ apples equally among themselves. We call $(x_A, x_B, x_C)$ a *stable configuration* if no child's total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of $n$ for which a stable configuration exists for some distribution $(a_1, a_2, \dots, a_n)$ of the apples.
Let $n$ be a positive integer and $\mathbb{R}^n$ be the set of ordered $n$-tuples of real numbers. Let $T$ denote the collection of $(x_1, x_2, \dots, x_n) \in \mathbb{R}^n$ for which there exists a permutation $\sigma$ of $1, 2, \dots, n$ such that $x_{\sigma(i)} - x_{\sigma(i+1)} \ge 1$ for each $1 \le i \lt n$. Prove that there is a real number $d$ satisfying the following condition:
For every $(a_1, a_2, \dots, a_n) \in \mathbb{R}^n$, there exist $(b_1, b_2, \dots, b_n)$, $(c_1, c_2, \dots, c_n) \in T$ such that
$$
a_i = \frac{1}{2}(b_i + c_i), \quad |a_i - b_i| \le d, \quad |a_i - c_i| \le d \quad (1 \le i \le n).
$$
In an acute triangle $ABC$ with $|BC| \lt |AC| \lt |AB|$, the points $D \in [AB]$ and $E \in [AC]$ satisfy the condition $|BD| = |BC| = |CE|$. Show that the circumradius of the triangle $ADE$ is equal to the distance between the incenter and the circumcenter of the triangle $ABC$.
Two circles are tangent to each other from outside at a point $A$ and to a third circle $\Gamma$ from inside at points $B$ and $C$. Let $D$ be the midpoint of the secant of $\Gamma$ which is tangent to the smaller circles at $A$. Show that $A$ is the incenter of the triangle $BCD$ if the centers of the circles are not collinear.
Graph Airlines (GA) operates flights between some of the cities of the Republic of Graphia. There are GA flights between each city and at least three different cities and it is possible to travel from any city to any other city in the Republic of Graphia using GA flights. GA decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel from any city to any other city using GA flights, yet at least $\frac{2}{9}$ of the cities has only one flight.
Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ satisfying the conditions $y^2 \equiv x^3 - x \pmod p$ and $0 \le x, y \lt p$ is exactly $p$.
How many ways are there of using diagonals to divide a regular $6$-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
Divide an $m$-by-$n$ rectangle into $mn$ nonoverlapping $1$-by-$1$ squares. A polyomino of this rectangle is a subset of these unit squares such that for any two unit squares $S, T$ in the polyomino, either
(1) $S$ and $T$ share an edge or
(2) there exists a positive integer $n$ such that the polyomino contains unit squares $S_{1}, S_{2}, S_{3}, \ldots, S_{n}$ such that $S$ and $S_{1}$ share an edge, $S_{n}$ and $T$ share an edge, and for all positive integers $k \lt n$, $S_{k}$ and $S_{k+1}$ share an edge.
We say a polyomino of a given rectangle spans the rectangle if for each of the four edges of the rectangle the polyomino contains a square whose edge lies on it.
What is the minimum number of unit squares a polyomino can have if it spans a $128$-by-$343$ rectangle?
Two circles have centers that are $d$ units apart, and each has diameter $\sqrt{d}$. For any $d$, let $A(d)$ be the area of the smallest circle that contains both of these circles. Find $\lim_{d \rightarrow \infty} \frac{A(d)}{d^{2}}$.
A math professor stands up in front of a room containing 100 very smart math students and says, "Each of you has to write down an integer between 0 and 100, inclusive, to guess 'two-thirds of the average of all the responses.' Each student who guesses the highest integer that is not higher than two-thirds of the average of all responses will receive a prize." If among all the students it is common knowledge that everyone will write down the best response, and there is no communication between students, what single integer should each of the 100 students write down?
How many ways, without taking order into consideration, can $2002$ be expressed as the sum of $3$ positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)?
Two integers are relatively prime if they don't share any common factors, i.e. if their greatest common divisor is $1$. Define $\varphi(n)$ as the number of positive integers that are less than $n$ and relatively prime to $n$. Define $\varphi_{d}(n)$ as the number of positive integers that are less than $d n$ and relatively prime to $n$.
What is the least $n$ such that $\varphi_{x}(n)=64000$, where $x=\varphi_{y}(n)$, where $y=\varphi(n)$ ?
Two $4 \times 4$ squares are randomly placed on an $8 \times 8$ chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap?
An $(l, a)$-design of a set is a collection of subsets of that set such that each subset contains exactly $l$ elements and that no two of the subsets share more than $a$ elements. How many $(2,1)$-designs are there of a set containing 8 elements?
Find the number of pentominoes (5-square polyominoes) that span a $3$-by-$3$ rectangle, where polyominoes that are flips or rotations of each other are considered the same polyomino.
given that $a, b, c$ are positive integers satisfying
$$
a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120
$$
determine the maximum possible value of $a$.