Found problems: 85335
1993 Taiwan National Olympiad, 1
A sequence $(a_{n})$ of positive integers is given by $a_{n}=[n+\sqrt{n}+\frac{1}{2}]$. Find all of positive integers which belong to the sequence.
2015 Purple Comet Problems, 27
A container is shaped like a right circular cone open at the top surmounted by a frustum which is open at
the top and bottom as shown below. The lower cone has a base with radius 2 centimeters and height 6
centimeters while the frustum has bases with radii 2 and 8 centimeters and height 6 centimeters. If there is
a rainfall measuring 2 centimeter of rain, the rain falling into the container will fill the container to a
height of $m + 3\sqrt{n}$ cm, where m and n are positive integers. Find m + n.
2025 CMIMC Team, 6
Suppose we have a regular $24$-gon labeled $A_1 \cdots A_{24}.$ We will draw $2$ similar $24$-gons within $A_1 \cdots A_{24}.$ For the sake of this problem, make $A_i=A_{i+24}.$
With our first configuration, we create $3$ stars by creating lines $\overline{A_iA_{i+9}}.$ A $24$-gon will be created in the center, which we denote as our first $24$-gon.
With our second configuration, we create a start by creating lines $\overline{A_iA_{i+11}}.$ A $24$-gon will be created in the center, which we denote as our second $24$-gon.
Find the ratio of the areas of the first $24$-gon to the second $24$-gon.
2006 Austrian-Polish Competition, 5
Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: \[\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}\]
2009 Iran MO (3rd Round), 3
An arbitary triangle is partitioned to some triangles homothetic with itself. The ratio of homothety of the triangles can be positive or negative.
Prove that sum of all homothety ratios equals to $1$.
Time allowed for this problem was 45 minutes.
1979 IMO Longlists, 14
Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds:
[b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty.
[b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.
2009 Chile National Olympiad, 4
Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$
1947 Putnam, A6
A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$
2016 China National Olympiad, 2
In $\triangle AEF$, let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$. Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively.
Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.
2022 Mid-Michigan MO, 5-6
[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm.
[b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions.
[img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img]
[b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles.
[img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img]
[b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number?
[b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct.
$$ 4 \times 12 + 18 : 6 + 3 = 50$$
[b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 Serbia National Math Olympiad, 2
Let $ABCD$ be a convex and cyclic quadrilateral.
Let $AD\cap BC=\{E\}$, and let $M,N$ be points on $AD,BC$ such that $AM:MD=BN:NC$. Circle around $\triangle EMN$ intersects circle around $ABCD$ at $X,Y$ prove that $AB,CD$ and $XY$ are either parallel or concurrent.
2017 Peru MO (ONEM), 2
Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board?
Clarification: Two squares are [i]neighbors [/i] if they have a common side.
1994 Irish Math Olympiad, 3
Find all real polynomials $ f(x)$ satisfying $ f(x^2)\equal{}f(x)f(x\minus{}1)$ for all $ x$.
1989 APMO, 4
Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least
\[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \]
triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.
2014 AMC 12/AHSME, 16
Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$?
$ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $
2020 CHMMC Winter (2020-21), 1
Triangle $ABC$ has circumcircle $\Omega$. Chord $XY$ of $\Omega$ intersects segment $AC$ at point $E$ and segment $AB$ at point $F$ such that $E$ lies between $X$ and $F$. Suppose that $A$ bisects arc $\widehat{XY}$. Given that $EC = 7, FB = 10, AF = 8$, and $YF - XE = 2$, find the perimeter of triangle $ABC$.
2025 India STEMS Category C, 1
Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$.
[i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]
2004 Cuba MO, 3
In the non-isosceles $\vartriangle ABC$, the interior bisectors of vertices $B$ and $C$ are drawn, which cut the sides $AC$ and $AB$ at $E$ and $F$ respectively.The line $EF$ cuts the extension of side $BC$ at $T$. In the side$ BC$ a point D is located, so that $\frac{DB}{DC} = \frac{TB}{TC}$. Prove that $AT$ is the exterior bisector of angle $A$.
2011 Preliminary Round - Switzerland, 1
Let $\triangle{ABC}$ a triangle with $\angle{CAB}=90^{\circ}$ and $L$ a point on the segment $BC$. The circumcircle of triangle $\triangle{ABL}$ intersects $AC$ at $M$ and the circumcircle of triangle $\triangle{CAL}$ intersects $AB$ at $N$. Show that $L$, $M$ and $N$ are collinear.
2013 ELMO Shortlist, 1
Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral.
[i]Proposed by Owen Goff[/i]
I Soros Olympiad 1994-95 (Rus + Ukr), 10.4
There are 1995 segments such that a triangle can be formed from any three of them. Prove that using these $1995 $ segments, it is possible to assemble $664$ acute-angled triangles so that each segment is part of no more than one triangle.
STEMS 2021 Math Cat C, Q3
Let $p \in \mathbb{N} \setminus \{0, 1\}$ be a fixed positive integer. Prove that for every $K > 0$, there exist infinitely many $n$ and $N$ such that there are atleast $\dfrac{KN}{\log(N)}$ primes among the following $N$ numbers given by
\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]
[i]Proposed by Bimit Mandal[/i]
1999 Nordic, 2
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.
1942 Putnam, A6
Any circle in the $xy$-plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle.
Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid.
By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.
2015 JBMO Shortlist, NT1
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?