Found problems: 85335
2014 Costa Rica - Final Round, 1
Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]
2014 South East Mathematical Olympiad, 7
Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.
2024 Stars of Mathematics, P3
Fix postive integer $n\geq 2$. Let $a_1,a_2,\dots ,a_n$ be real numbers in the interval $[1,2024]$. Prove that $$\sum_{i=1}^n\frac{1}{a_i}(a_1+a_2+\dots +a_i)>\frac{1}{44}n(n+33).$$
[i]Proposed by Radu-Andrei Lecoiu[/i]
2024 Chile Classification NMO Seniors, 2
Find all real numbers $x$ such that:
\[
2^x + 3^x + 6^x - 4^x - 9^x = 1,
\]
and prove that there are no others.
2020 Junior Macedonian National Olympiad, 3
Solve the following equation in the set of integers
$x^5 + 2 = 3 \cdot 101^y$.
2006 Tournament of Towns, 4
Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)
2022/2023 Tournament of Towns, P4
In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.
PEN N Problems, 3
Let $\,n>6\,$ be an integer and $\,a_{1},a_{2},\ldots,a_{k}\,$ be all the natural numbers less than $n$ and relatively prime to $n$. If \[a_{2}-a_{1}=a_{3}-a_{2}=\cdots =a_{k}-a_{k-1}>0,\] prove that $\,n\,$ must be either a prime number or a power of $\,2$.
1995 Bundeswettbewerb Mathematik, 2
A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$.
Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.
1952 AMC 12/AHSME, 47
In the set of equations $ z^x \equal{} y^{2x}, 2^z \equal{} 2\cdot4^x, x \plus{} y \plus{} z \equal{} 16$, the integral roots in the order $ x,y,z$ are:
$ \textbf{(A)}\ 3,4,9 \qquad\textbf{(B)}\ 9, \minus{} 5 \minus{} ,12 \qquad\textbf{(C)}\ 12, \minus{} 5,9 \qquad\textbf{(D)}\ 4,3,9 \qquad\textbf{(E)}\ 4,9,3$
2023 ELMO Shortlist, C5
Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list]
[i]Proposed by Espen Slettnes[/i]
Kvant 2020, M2599
Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways. (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot 2 \cdot . . . \cdot (2N)$.)
Gleb Pogudin
2018 Hanoi Open Mathematics Competitions, 8
Let $k$ be a positive integer such that $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.
2010 Today's Calculation Of Integral, 662
In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis.
(1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section.
(2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$.
(3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis.
[i]1987 Sophia University entrance exam/Science and Technology[/i]
2000 Harvard-MIT Mathematics Tournament, 43
Box A contains $3$ black and $4$ blue marbles. Box B has $7$ black and $1$ blue, whereas Box C has $2$ black, $3$ blue and $1$ green marble. I close my eyes and pick two marbles from $2$ different boxes. If it turns out that I get $1$ black and $1$ blue marble, what is the probability that the black marble is from box A and the blue one is from C?
2020 MMATHS, I6
Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors! How many possible values between $200$ and $1000$ are there for the total number of balls in the box?
[i]Proposed by Andrew Wu[/i]
2012 India IMO Training Camp, 1
Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.
2021 Sharygin Geometry Olympiad, 10-11.1
.Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.
2022 Bulgarian Autumn Math Competition, Problem 10.3
Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$
1975 Miklós Schweitzer, 11
Let $ X_1,X_2,...,X_n$ be (not necessary independent) discrete random variables. Prove that there exist at least $ n^2/2$ pairs
$ (i,j)$ such that \[ H(X_i\plus{}X_j) \geq \frac 13 \min_{1 \leq k \leq n} \{ H(X_k) \},\] where $ H(X)$ denotes the Shannon entropy of $ X$.
[i]GY. Katona[/i]
2022 HMNT, 33
A group of $101$ Dalmathians participate in an election, where they each vote independently on either candidate $A$ or $B$ with equal probability. If $X$ Dalmathians voted for the winning candidate, the expected value of $X^2$ can be expressed as $\tfrac{a}{b}$ for positive integers $a,b$ with $\gcd(a,b) = 1.$ Find the unique positive integer $k \le 103$ such that $103 | a-bk.$
OIFMAT I 2010, 4
Let $ a_1 <a_2 <... <a_n $ consecutive positive integers (with $ n> 2 $). A grasshopper jumps on the real line, starting at point $ 0 $ and jumping $ n $ to the right with lengths $ a_1 $, $ a_2 $, ..., $ a_n $, in some order (each length occupies exactly once), ending your tour at the $ 2010 $ point. Find all the possible values $ n $ of jumps that the grasshopper could have made.
2024 Mathematical Talent Reward Programme, 5
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that $f(m) - f(n) = f(m-n)10^n \forall m>n \in \mathbb{N}$. Additionally, gcd$(f(k),f(k+1)) = 1 \forall k \in \mathbb{N}$. Show that if $a,b$ are coprime natural numbers, that is, gcd$(a,b) = 1$ then $f(a),f(b)$ are also coprime.
2022 Baltic Way, 12
An acute-angled triangle $ABC$ has altitudes $AD, BE$ and $CF$. Let $Q$ be an interior point of the segment $AD$, and let the circumcircles of the triangles $QDF$ and $QDE$ meet the line $BC$ again at points $X$ and $Y$ , respectively. Prove that $BX = CY$ .
2003 Bosnia and Herzegovina Junior BMO TST, 3
Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.