Found problems: 85335
2009 Czech-Polish-Slovak Match, 5
The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following:
[list](i) $1\le a_1<a_2<\cdots < a_n\le 50$
(ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n. \] [/list]Prove that $n\le 16$ and determine the number of $n$-tuples $(a_1,a_2,\ldots,a_n$) satisfying these conditions for $n=16$.
2018 Stars of Mathematics, 4
Let be a natural number $ n\ge 4 $ and $ n $ nonnegative numbers $ a,b,\ldots ,c. $ Prove that
$$ \prod_{\text{cyc} } (a+b+c)^2 \ge 2^n\prod_{\text{cyc} } (a+b)^2, $$
and tell in which circumstances equality happens.
2015 India National Olympiad, 2
For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.
2016 Harvard-MIT Mathematics Tournament, 3
Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA = 90^{\circ}$. Given that $AI = 97$ and $BC = 144$, compute the area of $\triangle ABC$.
2009 Miklós Schweitzer, 6
A set system $ (S,L)$ is called a Steiner triple system, if $ L\neq\emptyset$, any pair $ x,y\in S$, $ x\neq y$ of points lie on a unique line $ \ell\in L$, and every line $ \ell\in L$ contains exactly three points. Let $ (S,L)$ be a Steiner triple system, and let us denote by $ xy$ the thrid point on a line determined by the points $ x\neq y$. Let $ A$ be a group whose factor by its center $ C(A)$ is of prime power order. Let $ f,h: S\to A$ be maps, such that $ C(A)$ contains the range of $ f$, and the range of $ h$ generates $ A$.
Show, that if
\[ f(x) \equal{} h(x)h(y)h(x)h(xy)\]
holds for all pairs $ x\neq y$ of points, then $ A$ is commutative, and there exists an element $ k\in A$, such that $ f(x) \equal{} kh(x)$ for all $ x\in S$.
2018 PUMaC Individual Finals B, 3
Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.
2005 Hungary-Israel Binational, 3
There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?
1993 Putnam, A5
Let U be the set formed as the union of three open intervals, $U = (-100, -10) \cup (1/101, 1/11) \cup (101/100, 11/10)$. Show that $\int_{U} \frac{(x^2-x)^2}{(x^3-3x+1)^2} dx$ is rational.
2013 CHMMC (Fall), 3
Bill plays a game in which he rolls two fair standard six-sided dice with sides labeled one through six. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once.
Russian TST 2016, P2
In a class, there are $n{}$ children of different heights. Denote by $A{}$ the number of ways to arrange them all in a row, numbered $1,2,\ldots,n$ from left to right, so that each person with an odd number is shorter than each of his neighbors. Let $B{}$ be the number of ways to organize $n-1$ badminton games between these children so that everyone plays at most two games with children shorter than himself and at most one game with children taller than himself (the order of the games is not important). Prove that $A = B$.
2015 USA TSTST, 3
Let $P$ be the set of all primes, and let $M$ be a non-empty subset of $P$. Suppose that for any non-empty subset ${p_1,p_2,...,p_k}$ of $M$, all prime factors of $p_1p_2...p_k+1$ are also in $M$. Prove that $M=P$.
[i]Proposed by Alex Zhai[/i]
2020 Turkey EGMO TST, 2
$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$
2019 AMC 12/AHSME, 4
What is the greatest number of consecutive integers whose sum is $45 ?$
$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$
2014 JBMO TST - Turkey, 2
Find all triples of positive integers $(a, b, c)$ satisfying $(a^3+b)(b^3+a)=2^c$.
2019 Flanders Math Olympiad, 1
Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$
[img]https://1.bp.blogspot.com/-O4B3P3bghFs/Xy1fDv9zGkI/AAAAAAAAMSQ/ePLVnsXsRi0mz3SWBpIzfGdsizWoLmGVACLcBGAsYHQ/s0/flanders%2B2019%2Bp1.png[/img]
2008 Costa Rica - Final Round, 3
Find all polinomials $ P(x)$ with real coefficients, such that
$ P(\sqrt {3}(a \minus{} b)) \plus{} P(\sqrt {3}(b \minus{} c)) \plus{} P(\sqrt {3}(c \minus{} a)) \equal{} P(2a \minus{} b \minus{} c) \plus{} P( \minus{} a \plus{} 2b \minus{} c) \plus{} P( \minus{} a \minus{} b \plus{} 2c)$
for any $ a$,$ b$ and $ c$ real numbers
2011 Puerto Rico Team Selection Test, 6
Find all natural numbers such that each is equal to the sum of the factorials of its digits.
Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
2016 Kyrgyzstan National Olympiad, 2
The number $N$ consists only $2's$ and $1's$ in its [b]decimal representation[/b].We know that,after deleting digits from N,we can get any number consisting $9999$- $1's$ and $one$ - $2's$ in its [b]decimal representation[/b].[b][u]Find the least number of digits in the decimal representation of N[/u][/b]
2000 Polish MO Finals, 2
In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$x$n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even
2018 China Team Selection Test, 3
Circle $\omega$ is tangent to sides $AB$,$AC$ of triangle $ABC$ at $D$,$E$ respectively, such that $D\neq B$, $E\neq C$ and $BD+CE<BC$. $F$,$G$ lies on $BC$ such that $BF=BD$, $CG=CE$. Let $DG$ and $EF$ meet at $K$. $L$ lies on minor arc $DE$ of $\omega$, such that the tangent of $L$ to $\omega$ is parallel to $BC$. Prove that the incenter of $\triangle ABC$ lies on $KL$.
2024 AMC 10, 21
Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(1.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]
$\textbf{(A)}~\displaystyle\frac{1}{9}
\qquad\textbf{(B)}~1
\qquad\textbf{(C)}~\displaystyle\frac{10}{9}
\qquad\textbf{(D)}~\displaystyle\frac{11}{9}
\qquad\textbf{(E)}~\displaystyle\frac{19}{9}$
2017 AMC 10, 9
A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
$\textbf{(A) } \frac{1}{27}\qquad \textbf{(B) } \frac{1}{9}\qquad \textbf{(C) } \frac{2}{9}\qquad \textbf{(D) } \frac{7}{27}\qquad \textbf{(E) } \frac{1}{2}$
2008 JBMO Shortlist, 9
Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the sequence $1, 2, 3, ..., 1024 $ the operation is performed successively for $5$ times. Show that at the end only one number remains and fi nd this number.
2025 CMIMC Combo/CS, 7
Alan is bored one day and decides to write down all the divisors of $1260^2$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to erase all his work. Every second, he picks a random divisor which is still on the wall and instantly erases it and every number that divides it. What is the expected time it takes for Alan to erase everything on the wall?
1993 National High School Mathematics League, 15
Positive sequence $(a_n)_{n=0}^{\infty}$ satisfies that $\sqrt{a_na_{n-2}}-\sqrt{a_{n-1}a_{n-2}}=2a_{n-1}(n\geq2),a_0=a_1=1$. Find $a_n$.