Found problems: 85335
2012 CHMMC Spring, 2
A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\sqrt{17}$, $\sqrt{37}$, $\sqrt{52}$. A second triangular face $F_0$ has side lengths $\sqrt{13}$, $\sqrt{29}$, $\sqrt{34}$. What is the minimum possible volume of the octahedron?
2016 Japan Mathematical Olympiad Preliminary, 10
Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this.
Note that boy A doesn’t have to return to the starting point to leave gotten flags.
2023 Malaysian IMO Training Camp, 1
Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors?
[i]Proposed by Wong Jer Ren[/i]
Kyiv City MO Seniors 2003+ geometry, 2011.10.3
A trapezoid $ABCD$ with bases $BC = a$ and $AD = 2a$ is drawn on the plane. Using only with a ruler, construct a triangle whose area is equal to the area of the trapezoid. With the help of a ruler you can draw straight lines through two known points.
(Rozhkova Maria)
2020 BMT Fall, 18
Let $T$ be the answer to question $17$, and let $N =\frac{24}{T}$. Leanne flips a fair coin $N$ times. Let $X$ be the number of times that within a series of three consecutive flips, there were exactly two heads or two tails. What is the expected value of $X$?
1952 AMC 12/AHSME, 4
The cost $ C$ of sending a parcel post package weighing $ P$ pounds, $ P$ and integer, is $ 10$ cents for the first pound and $ 3$ cents for each additional pound. The formula for the cost is:
$ \textbf{(A)}\ C \equal{} 10 \plus{} 3P \qquad\textbf{(B)}\ C \equal{} 10P \plus{} 3 \qquad\textbf{(C)}\ C \equal{} 10 \plus{} 3(P \minus{} 1)$
$ \textbf{(D)}\ C \equal{} 9 \plus{} 3P \qquad\textbf{(E)}\ C \equal{} 10P \minus{} 7$
2014 Contests, 1
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$
2004 VJIMC, Problem 3
Let $\sum_{n=1}^\infty a_n$ be a divergent series with positive nonincreasing terms. Prove that the series
$$\sum_{n=1}^\infty\frac{a_n}{1+na_n}$$diverges.
2013 AIME Problems, 9
A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
[asy]
import cse5;
size(12cm);
pen tpen = defaultpen + 1.337;
real a = 39/5.0;
real b = 39/7.0;
pair B = MP("B", (0,0), dir(200));
pair A = MP("A", (9,0), dir(-80));
pair C = MP("C", (12,0), dir(-20));
pair K = (6,10.392);
pair M = (a*B+(12-a)*K) / 12;
pair N = (b*C+(12-b)*K) / 12;
draw(B--M--N--C--cycle, tpen);
draw(M--A--N--cycle);
fill(M--A--N--cycle, mediumgrey);
pair shift = (-20.13, 0);
pair B1 = MP("B", B+shift, dir(200));
pair A1 = MP("A", K+shift, dir(90));
pair C1 = MP("C", C+shift, dir(-20));
draw(A1--B1--C1--cycle, tpen);[/asy]
2018 China Team Selection Test, 4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
Math Hour Olympiad, Grades 5-7, 2019.67
[u]Round 1[/u]
[b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$?
[b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated?
[img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img]
[b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time.
[img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img]
[b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken.
[img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img]
[b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$?
[u]Round 2[/u]
[b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img]
Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one?
[img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img]
[b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2013 ELMO Shortlist, 3
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.
Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.
[i]Proposed by Matthew Babbitt[/i]
2019 IFYM, Sozopol, 3
The natural number $n>1$ is such that there exist $a\in \mathbb{N}$ and a prime number $q$ which satisfy the following conditions:
1) $q$ divides $n-1$ and $q>\sqrt{n}-1$
2) $n$ divides $a^{n-1}-1$
3) $gcd(a^\frac{n-1}{q}-1,n)=1$.
Is it possible for $n$ to be a composite number?
Kvant 2024, M2806
Is it possible to draw a closed $20$-link polyline on the plane and number its links with the numbers $1, 2, 3, \ldots, 20$ in the order of traversal so that for each natural $i = 1, 2, 3, \ldots, 10$ the links numbered $i$ and $10+i$ intersect each other and do not intersect the other links?
[i] I. Efremov[/i]
2014 France Team Selection Test, 4
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
2016 Brazil Team Selection Test, 2
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions.
PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016
2000 Estonia National Olympiad, 4
Prove that for any triangle the equation holds $a \cdot \cos (\beta + \gamma ) + b \cdot \cos (\gamma +\alpha) + c\cdot \cos (\alpha -\beta) = 0$,
where $a, b, c$ are the sides of the triangle and $\alpha, \beta,\gamma$ according to their angles sizes of opposite angles.
2019 Saudi Arabia JBMO TST, 3
Find all positive integers of form abcd such that
$$\overline{abcd} = a^{a+b+c+d} - a^{-a+b-c+d} + a$$
2008 Vietnam National Olympiad, 5
What is the total number of natural numbes divisible by 9 the number of digits of which does not exceed 2008 and at least two of the digits are 9s?
2024 Iran MO (2nd Round), 3
In a triangle $ABC$ the incenter, the $B$-excenter and the $C$-excenter are $I, K$ and $L$, respectively. The perpendiculars at $B$ and $C$ to $BC$ intersect the lines $AC$ and $AB$ at $E$ and $F$, respectively. Prove that the circumcircles of $AEF, FIL, EIK$ concur.
2020 Caucasus Mathematical Olympiad, 4
Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.
2010 Contests, 3
Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.
2000 All-Russian Olympiad Regional Round, 11.2
The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?
LMT Accuracy Rounds, 2023 S4
Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .
1994 National High School Mathematics League, 6
In rectangular coordinate system, the equation $\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1$ ($a,b$ are different positive numbers) refers to
$\text{(A)}$ a triangle
$\text{(B)}$ a square
$\text{(C)}$ rectangle, not square
$\text{(D)}$ rhombus, not square