In an acute-angled triangle $ABC$, we consider the feet $H_a$ and $H_b$ of the altitudes from $A$ and $B$, and the intersections $W_a$ and $W_b$ of the angle bisectors from $A$ and $B$ with the opposite sides $BC$ and $CA$ respectively. Show that the centre of the incircle $I$ of triangle $ABC$ lies on the segment $H_aH_b$ if and only if the centre of the circumcircle $O$ of triangle $ABC$ lies on the segment $W_aW_b$.

Prove that the number (a) $97^{97}$ (b) $1997^{17}$ cannot be equal to a sum of cubes of several consecutive integers. (AA Egorov)

Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.

Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.

Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

Two equal circles $\Omega_1$ and $\Omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Two rays were drawn from $M$, lying in the same half-plane wrt $AB$ (see figure). The first ray intersects the circles $\Omega_1$ and $\Omega_2$ at points $X_1$ and $X_2$, and the second ray intersects them at points $Y_1$ and $Y_2$, respectively. Let $C$ be the intersection point of straight lines $AX_1$ and $BY_2$, and let $D$ be the intersection point of straight lines $AX_2$ and $BY_1$. Prove that $CD \parallel AB$. [img]https://cdn.artofproblemsolving.com/attachments/4/a/fae047c3956d8b30f15a9d88e8d12e5f4d48ec.png[/img]

The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

A teacher wishes to separate her $12$ students into groups. Yesterday, the teacher put the students into $4$ groups of $3$. Today, the teacher decides to put the students into $4$ groups of $3$ again. However, she doesn’t want any pair of students to be in the same group on both days. Find how many ways she could formthe groups today.

Let $ABCD$ be a rectangle with $AB=8$ and $AD=20$. Two circles of radius $5$ are drawn with centers in the interior of the rectangle - one tangent to $AB$ and $AD$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles?

In the country of Muilejistan, there exists a network of roads connecting all its cities. The network has the particular property that for any two cities, there is a unique path without backtracking (i.e., a path where the traveler never returns along the same road). The longest possible path between two cities is 600 kilometers. For instance, the path from the city of Mlar to the city of Nlar is 600 kilometers. Similarly, the path from the city of Klar to the city of Glar is also 600 kilometers. 1. If Jalim departs from Mlar towards Nlar at noon and Kalim departs from Klar towards Glar also at noon, both traveling at the same speed, prove that they meet at some point on their journey. 2. If the distance in kilometers between any two cities is an integer, prove that the distance from Glar to Mlar is even.

Let $n$ be a positive integer. Is it possible to express $n^2+3n+3$ into the form $ab$ with $a$ and $b$ being positive integers, and such that the difference between $a$ and $b$ is smaller than $2\sqrt{n+1}$?

Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$. Proposed by Ephram Chun