Found problems: 85335
2022 CCA Math Bonanza, L5.4
Five points are selected within a unit circle at random. Estimate the minimum distance between any pair of points. An estimate $E$ earns $\frac{2}{1+|log_2(A)-log_2(E)|}$ points, where $A$ is the actual answer.
[i]2022 CCA Math Bonanza Lightning Round 5.4[/i]
1971 Polish MO Finals, 3
A safe is protected with a number of locks. Eleven members of the committee have keys for some of the locks. What is the smallest number of locks necessary so that every six members of the committee can open the safe, but no five members can do it? How should the keys be distributed among the committee members if the number of locks is the smallest?
2014 CentroAmerican, 1
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
1995 IMO Shortlist, 6
Let $ A_1A_2A_3A_4$ be a tetrahedron, $ G$ its centroid, and $ A'_1, A'_2, A'_3,$ and $ A'_4$ the points where the circumsphere of $ A_1A_2A_3A_4$ intersects $ GA_1,GA_2,GA_3,$ and $ GA_4,$ respectively. Prove that
\[ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4\]
and
\[ \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.\]
2014 HMNT, 5
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
(By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)
2014 NIMO Problems, 1
Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
[i]Proposed by Kevin Sun[/i]
1998 Denmark MO - Mohr Contest, 1
In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure.
[img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]
2008 Rioplatense Mathematical Olympiad, Level 3, 1
Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?
2006 MOP Homework, 5
Show that among the vertices of any area $1$ convex polygon with $n > 3$ sides there exist four such that the quadrilateral formed by these four has area at least $1/2$.
2021 Philippine MO, 5
A positive integer is called $\emph{lucky}$ if it is divisible by $7$, and the sum of its digits is also divisible by $7$. Fix a positive integer $n$. Show that there exists some lucky integer $l$ such that $\left|n - l\right| \leq 70$.
2025 NCMO, 2
In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent.
[i]Alan Cheng[/i]
2021 Latvia TST, 2.2
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
2019 Durer Math Competition Finals, 8
A chess piece is placed on one of the squares of an $8\times 8$ chessboard where it begins a tour of the board: it moves from square to square, only moving horizontally or vertically. It visits every square precisely once, and ends up exactly where it started. What is the maximum number of times the piece can change direction along its tour?
2007 Moldova Team Selection Test, 4
Consider a convex polygon $A_{1}A_{2}\ldots A_{n}$ and a point $M$ inside it. The lines $A_{i}M$ intersect the perimeter of the polygon second time in the points $B_{i}$. The polygon is called balanced if all sides of the polygon contain exactly one of points $B_{i}$ (strictly inside). Find all balanced polygons.
[Note: The problem originally asked for which $n$ all convex polygons of $n$ sides are balanced. A misunderstanding made this version of the problem appear at the contest]
2017 Saudi Arabia IMO TST, 1
Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \in AC, F \in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\angle OKT = 90^o$.
2005 Italy TST, 2
$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality.
$(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.
2021 AMC 10 Fall, 9
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{3}{13}\qquad\textbf{(C) }\frac{7}{27}\qquad\textbf{(D) }\frac{2}{7}\qquad\textbf{(E) }\frac{1}{3}$
1998 All-Russian Olympiad Regional Round, 8.3
There are 52 cards in the deck, 13 of each suit. Vanya draws from the deck one card at a time. Removed cards are not returned to the deck. Every time Before taking out the card, Vanya makes a wish for some suit.Prove that if Vanya makes a wish every time, , the cards of which are in the deck has no less cards left than cards of any other suit, then the hidden suit will fall with the suit of the card drawn at least 13 times.
1966 IMO Longlists, 12
Find digits $x, y, z$ such that the equality
\[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\]
holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.
2009 Germany Team Selection Test, 3
There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that
\[
\angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ}
\]
if and only if the diagonals $ AC$ and $ BD$ are perpendicular.
[i]Proposed by Dusan Djukic, Serbia[/i]
1985 Federal Competition For Advanced Students, P2, 4
Find all natural numbers $ n$ such that the equation:
$ a_{n\plus{}1} x^2\minus{}2x \sqrt{a_1^2\plus{}a_2^2\plus{}...\plus{}a_{n\plus{}1}^2}\plus{}a_1\plus{}a_2\plus{}...\plus{}a_n\equal{}0$
has real solutions for all real numbers $ a_1,a_2,...,a_{n\plus{}1}$.
2016 Thailand TSTST, 2
Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.
1997 Pre-Preparation Course Examination, 1
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
2021 AMC 12/AHSME Fall, 17
A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?
$\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\
\frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}$
2015 USAMO, 6
Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)