At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?

Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.

$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.

Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $? $\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $ [asy] pair A,B,C,D,E; A=(0,0); B=(0,50); C=(50,50); D=(50,0); E = (30,50); draw(A--B); draw(B--E); draw(E--C); draw(C--D); draw(D--A); draw(A--E); dot(A); dot(B); dot(C); dot(D); dot(E); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,N); [/asy]

Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that - for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and - the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$ Prove that \[{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.\]

For which natural numbers $ n $ the floor of the number $ \frac{n^3+8n^2+1}{3n} $ is prime? [i]Gabriel Popa[/i]

The following list shows every number for which more than half of its digits are digits $2$, in increasing order: $$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$ If the $n$th term in the list is $2022$, what is $n$?

Suppose that $a,b,c,d$ are non-negative real numbers such that $a^2+b^2+c^2+d^2=2$ and $ab+bc+cd+da=1$. Find the maximum value of $a+b+c+d$ and determine all equality cases.

Original in Hungarian; translated with Google translate; polished by myself. Let $n$ be a positive integer. Suppose that the sum of the matrices $A_1, \dots, A_n\in \Bbb R^{n\times n}$ is the identity matrix, but $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is singular whenever at least one of the coefficients $\alpha_i \in \Bbb R$ is zero. a) Show that $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is nonsingular if $\alpha_i\neq 0$ for all $i$. b) Show that if the matrices $A_i$ are symmetric, then all of them have rank $1$.

Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.

For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.

Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.

Given $R, r > 0$. Two circles are drawn radius $R$, $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$?

Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac{5}{2} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{7}{2} \qquad \textbf{(E)}\ 4$

A policeman is trying to catch a robber on a board of $2001\times2001$ squares. They play alternately, and the player whose trun it is moves to a space in one of the following directions: $\downarrow,\rightarrow,\nwarrow$. If the policeman is on the square in the bottom-right corner, he can go directly to the square in the upper-left corner (the robber can not do this). Initially the policeman is in the central square, and the robber is in the upper-left adjacent square. Show that: $a)$ The robber may move at least $10000$ times before the being captured. $b)$ The policeman has an strategy such that he will eventually catch the robber. Note: The policeman can catch the robber if he reaches the square where the robber is, but not if the robber enters the square occupied by the policeman.

Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and \[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \] for all integers \(k \geq 1\). Determine all positive integers \(n\) such that \[ \frac{a_n}{n} \] is an integer. Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.

Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$, respectively, are externally tangent at point $\lambda$. A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$, $O_2$ lie inside $C$. The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$. If $D$ is the midpoint of the segment $KL$, show that $\angle O_1OO_2 = \angle ADB$. [i]Greece[/i]

Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.

In a virtually-made world, each citizen, which is assumed to be a point (i.e. without area) and labelled as $1, 2, ...$. To fight against a pandemic, these citizens are required to get vaccinated. After they get vaccinated, they need to be observed for a period of time. Now assume the location that the citizens get observe is a circumference with radius $\frac{1}{4}$ on the plane. For the safety reason, it is required for distance between $m$-th citizen and $n$-th citizen $d_{m, n}$ satisfying the following: $(m+n)d_{m, n}\geq 1$ Here what we consider is the distance on the circumference i.e. the arc length of minor arc formed by two points. Then (a) Choose one of the following which fits the situation in reality. A. The location for observation can mostly have $8$ citizens. B. The location for observation can have the upper limit on the number of citizens which is larger than $8$. C. The location for observation can have any number of citizens. (b) Prove your answer in (a).

$(a)$ Is it possible to partition the segment $[0,1]$ into two sets $A$ and $B$ and to define a continuous function $f$ such that for every $x\in A \ f(x)$ is in $B$, and for every $x\in B \ f(x)$ is in $A$? $(b)$ The same question with $[0,1]$ replaced by $[0,1).$

The polygon $P$, cut out of paper, is bent in a straight line and both halves are glued. Can the perimeter of the polygon $Q$ obtained by gluing be larger than the perimeter of the polygon $P$?

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

Find all positive integers $a,n\ge1$ such that for all primes $p$ dividing $a^n-1$, there exists a positive integer $m<n$ such that $p\mid a^m-1$.