Found problems: 85335
1989 Iran MO (2nd round), 1
[b](a)[/b] Let $n$ be a positive integer, prove that
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\]
[b](b)[/b] Find a positive integer $n$ for which
\[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]
2017 Brazil Team Selection Test, 1
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2023 ELMO Shortlist, C7
A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\).
Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed.
[i]Proposed by Linus Tang[/i]
1955 Putnam, A2
$A_1 ~A_2~ \ldots ~A_n$ is a regular polygon inscribed in a circle of radius $r$ and center $O.$ $P$ is a point on line $OA_1$ extended beyond $A_1.$ Show that \[ \prod^n_{i=1} ~ \overline{PA}_{~i} = \overline{OP}^{~n} - r^n. \]
Geometry Mathley 2011-12, 8.1
Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$.
Kostas Vittas
1966 Swedish Mathematical Competition, 2
$a_1 + a_2 + ... + a_n = 0$, for some $k$ we have $a_j \le 0$ for $j \le k$ and $a_j \ge 0$ for $j > k$. If ai are not all $0$, show that $a_1 + 2a_2 + 3a_3 + ... + na_n > 0$.
2011 Harvard-MIT Mathematics Tournament, 2
Let $a \star b = ab + a + b$ for all integers $a$ and $b$. Evaluate $1 \star ( 2 \star ( 3 \star (4 \star \ldots ( 99 \star 100 ) \ldots )))$.
2024 CMIMC Combinatorics and Computer Science, 1
For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. For how many three-digit positive integers $m$ is $m + t(t(m))$ odd?
[i]Proposed by David Altizio[/i]
2016 NIMO Problems, 1
Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$.
[i]Proposed by Michael Tang[/i]
2015 ASDAN Math Tournament, 36
A blue square of side length $10$ is laid on top of a coordinate grid with corners at $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$. Red squares of side length $2$ are randomly placed on top of the grid, changing the color of a $2\times2$ square section red. Each red square when placed lies completely within the blue square, and each square's four corners take on integral coordinates. In addition, randomly placed red squares may overlap, keeping overlapped regions red. Compute the expected value of the number of red squares necessary to turn the entire blue square red, rounded to the nearest integer. Your score will be given by $\lfloor25\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.
2017 CCA Math Bonanza, TB1
Compute \[12^3+4\times56+7\times8+9.\]
[i]2017 CCA Math Bonanza Tiebreaker Round #1[/i]
2013 All-Russian Olympiad, 1
$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values of their pairwise differences, there are ten different numbers not exceeding $100$.
2018 Ecuador Juniors, 2
Danielle divides a $30 \times30$ board into $100$ regions that are $3 \times 3$ squares squares each and then paint some squares black and the rest white. Then to each region assigns it the color that has the most squares painted with that color.
a) If there are more black regions than white, what is the minimum number $N$ of cells that Danielle can paint black?
b) In how many ways can Danielle paint the board if there are more black regions than white and she uses the minimum number $N$ of black squares?
1996 Greece National Olympiad, 4
Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.
2009 Greece Team Selection Test, 2
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
1983 IMO Longlists, 69
Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.
1997 Bulgaria National Olympiad, 3
Let $n$ and $m$ be natural numbers such that $m+ i=a_ib_i^2$ for $i=1,2, \cdots n$ where $a_i$ and $b_i$ are natural numbers and $a_i$ is not divisible by a square of a prime number.
Find all $n$ for which there exists an $m$ such that
$\sum_{i=1}^{n}a_i=12$
2004 USAMTS Problems, 1
Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers.
2012 NZMOC Camp Selection Problems, 2
Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.
2002 Iran Team Selection Test, 1
$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.
2023 SG Originals, Q3
Let $n \geq 2$ be a positive integer. For a positive integer $a$, let $Q_a(x)=x^n+ax$. Let $p$ be a prime and let $S_a=\{b | 0 \leq b \leq p-1, \exists c \in \mathbb {Z}, Q_a(c) \equiv b \pmod p \}$. Show that $\frac{1}{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.
PEN H Problems, 24
Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.
2019 IMO, 5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.
[i]Proposed by David Altizio, USA[/i]
1935 Moscow Mathematical Olympiad, 006
The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.
2024 Caucasus Mathematical Olympiad, 6
Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold:
1) their sum is $a$;
2) the absolute value of their difference is $b$.
Find all possible values of $b$.