Found problems: 85335
2014 IMO, 2
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
2023 Hong Kong Team Selection Test, Problem 2
Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the [i]repetend[/i] of the fraction, and the smallest length of the repetend, $k$, is called the [i]period[/i] of the decimal number.)
2018 Macedonia JBMO TST, 4
Determine all pairs $(p, q)$, $p, q \in \mathbb {N}$, such that
$(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q$.
2008 239 Open Mathematical Olympiad, 3
Prove that you can arrange arrows on the edges of a convex polyhedron such that each vertex contains at most three arrows.
1987 IMO Longlists, 15
Let $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ be nine strictly positive real numbers. We set
\[S_1 = a_1b_2c_3, \quad S_2 = a_2b_3c_1, \quad S_3 = a_3b_1c_2;\]\[T_1 = a_1b_3c_2, \quad T_2 = a_2b_1c_3, \quad T_3 = a_3b_2c_1.\]
Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements.
Prove that
\[S_1 + S_2 + S_3 = T_1 + T_2 + T_3.\]
2017 Polish Junior Math Olympiad Second Round, 2.
Prove that if the diagonals of a certain trapezoid are perpendicular, then the sum of the lengths of the bases of this trapezoid is not greater than the sum of the lengths of the sides of this trapezoid.
2019 Korea - Final Round, 3
Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying
$$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$
does not contain any prime number.
1990 Baltic Way, 6
Let $ABCD$ be a quadrilateral with $AD = BC$ and $\angle DAB + \angle ABC = 120^\circ$. An equilateral triangle $DPC$ is erected in the exterior of the quadrilateral. Prove that the triangle $APB$ is also equilateral.
2015 CCA Math Bonanza, T8
Triangle $ABC$ is equilateral with side length $\sqrt{3}$ and circumcenter at $O$. Point $P$ is in the plane such that $(AP)(BP)(CP) = 7$. Compute the difference between the maximum and minimum possible values of $OP$.
[i]2015 CCA Math Bonanza Team Round #8[/i]
2011 AMC 12/AHSME, 22
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\ 1500 \qquad
\textbf{(B)}\ 1560 \qquad
\textbf{(C)}\ 2320 \qquad
\textbf{(D)}\ 2480 \qquad
\textbf{(E)}\ 2500$
1972 All Soviet Union Mathematical Olympiad, 168
A game for two.
One gives a digit and the second substitutes it instead of a star in the following difference:
$$**** - **** = $$
Then the first gives the next digit, and so on $8$ times.
The first wants to obtain the greatest possible difference, the second -- the least. Prove that:
1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour.
2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.
2010 Putnam, A5
Let $G$ be a group, with operation $*$. Suppose that
(i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
(ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
1963 Swedish Mathematical Competition., 4
Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.
2019 AMC 10, 22
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$?
$\textbf{(A)} \frac{1}{3} \qquad \textbf{(B)} \frac{7}{16} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{9}{16} \qquad \textbf{(E)} \frac{2}{3}$
2013 Today's Calculation Of Integral, 898
Let $a,\ b$ be positive constants.
Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]
2024 Rioplatense Mathematical Olympiad, 5
Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy
\[
\text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab)
\]
for all pairs of integers $a, b \in S$.
Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.
2020/2021 Tournament of Towns, P2
Let $AX$ and $BZ$ be altitudes of the triangle $ABC$. Let $AY$ and $BT$ be its angle bisectors. It is given that angles $XAY$ and $ZBT$ are equal. Does this necessarily imply that $ABC$ is isosceles?
[i]The Jury[/i]
2009 Romania National Olympiad, 4
We say that a natural number $ n\ge 4 $ is [i]unusual[/i] if, for any $ n\times n $ array of real numbers, the sum of the numbers from any $ 3\times 3 $ compact subarray is negative, and the sum of the numbers from any $ 4\times 4 $ compact subarray is positive.
Find all unusual numbers.
2014 NIMO Problems, 1
Compute $1+2\cdot3^4$.
[i]Proposed by Evan Chen[/i]
2008 IMO Shortlist, 6
Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$.
[i]Author: Kestutis Cesnavicius, Lithuania[/i]
2005 AMC 8, 24
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$
2013 Princeton University Math Competition, 2
Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.
1976 AMC 12/AHSME, 1
If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals
$\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }1/2\qquad\textbf{(D) }2\qquad \textbf{(E) }3$
2024 Malaysian IMO Training Camp, 1
Let $ABC$ be an acute triangle with orthocenter $H$, and let $BE$ and $CF$ be the altitudes of the triangle. Choose two points $P$ and $Q$ on rays $BH$ and $CH$ respectively, such that:
$\bullet$ $PQ$ is parallel to $BC$;
$\bullet$ The quadrilateral $APHQ$ is cyclic.
Suppose the circumcircles of triangles $APF$ and $AQE$ meet again at $X\neq A$. Prove that $AX$ is parallel to $BC$.
[i]Proposed by Ivan Chan Kai Chin[/i]
1968 Miklós Schweitzer, 3
Let $ K$ be a compact topological group, and let $ F$ be a set of continuous functions defined on $ K$ that has cardinality greater that continuum. Prove that there exist $ x_0 \in K$ and $ f \not\equal{}g \in F$ such that
\[ f(x_0)\equal{}g(x_0)\equal{}\max_{x\in K}f(x)\equal{}\max_{x \in K}g(x).\]
[i]I. Juhasz[/i]