Found problems: 85335
1990 IMO Longlists, 31
Let $S = \{1, 2, \ldots, 1990\}$. A $31$-element subset of $S$ is called "good" if the sum of its elements is divisible by $5$. Find the number of good subsets of $S.$
2012 Czech-Polish-Slovak Match, 3
Let $a,b,c,d$ be positive real numbers such that $abcd=4$ and
\[a^2+b^2+c^2+d^2=10.\]
Find the maximum possible value of $ab+bc+cd+da$.
1985 Traian Lălescu, 1.4
Let $ a $ be a non-negative real number distinct from $ 1. $
[b]a)[/b] For which positive values $ x $ the equation
$$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$
is true?
[b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $
1999 Portugal MO, 3
If two parallel chords of a circumference, $10$ mm and $14$ mm long, with distance $6$ mm from each other, how long is the chord equidistant from these two?
1994 Vietnam Team Selection Test, 3
Calculate
\[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\]
where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$
2007 Paraguay Mathematical Olympiad, 3
Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$.
a) Show that $DE \perp CF$.
b) Determine the ratio $CF : PC : EP$
2004 Iran MO (3rd Round), 5
assume that k,n are two positive integer $k\leq n$count the number of permutation $\{\ 1,\dots ,n\}\ $ st for any $1\leq i,j\leq k$and any positive integer m we have $f^m(i)\neq j$ ($f^m$ meas iterarte function,)
2000 China National Olympiad, 2
A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$,
\[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\]
Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.
2024 Francophone Mathematical Olympiad, 1
Let $d$ and $m$ be two fixed positive integers. Pinocchio and Geppetto know the values of $d$ and $m$ and play the following game: In the beginning, Pinocchio chooses a polynomial $P$ of degree at most $d$ with integer coefficients. Then Geppetto asks him questions of the following form "What is the value of $P(n)$?'' for $n \in \mathbb{Z}$. Pinocchio usually says the truth, but he can lie up to $m$ times. What is, as a function of $d$ and $m$, the minimal number of questions that Geppetto needs to ask to be sure to determine $P$, no matter how Pinocchio chooses to reply?
2013 CHMMC (Fall), 2
Two circles of radii $7$ and $17$ have a distance of $25$ between their centers. What is the difference between the lengths of their common internal and external tangents (positive difference)?
2014 ASDAN Math Tournament, 9
A sequence $\{a_n\}_{n\geq0}$ obeys the recurrence $a_n=1+a_{n-1}+\alpha a_{n-2}$ for all $n\geq2$ and for some $\alpha>0$. Given that $a_0=1$ and $a_1=2$, compute the value of $\alpha$ for which
$$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$
2019 Junior Balkan Team Selection Tests - Moldova, 3
Let $O$ be the center of circumscribed circle $\Omega$ of acute triangle $\Delta ABC$. The line $AC$ intersects the circumscribed circle of triangle $\Delta ABO$ for the second time in $X$. Prove that $BC$ and $XO$ are perpendicular.
2015 VTRMC, Problem 3
Let $(a_i)_{1\le i\le2015}$ be a sequence consisting of $2015$ integers, and let $(k_i)_{1\le i\le2015}$ be a sequence of $2015$ positive integers (positive integer excludes $0$). Let
$$A=\begin{pmatrix}a_1^{k_1}&a_1^{k_2}&\cdots&a_1^{k_{2015}}\\a_2^{k_1}&a_2^{k_2}&\cdots&a_2^{k_{2015}}\\\vdots&\vdots&\ddots&\vdots\\a_{2015}^{k_1}&a_{2015}^{k_2}&\cdots&a_{2015}^{k_{2015}}\end{pmatrix}.$$Prove that $2015!$ divides $\det A$.
2023 Kyiv City MO Round 1, Problem 1
Which number is larger: $A = \frac{1}{9} : \sqrt[3]{\frac{1}{2023}}$, or $B = \log_{2023} 91125$?
2010 All-Russian Olympiad Regional Round, 9.7
In a company of seven people, any six can sit at a round table so that every two neighbors turn out to be acquaintances. Prove that the whole company can be seated at a round table so that every two neighbors turn out to be acquaintances.
2010 LMT, 8
The integer $111111$ is the product of five prime numbers. Determine the sum of these prime numbers.
1966 IMO Longlists, 31
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?
2007 AIME Problems, 10
Let $S$ be a set with six elements. Let $P$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $P$. the probability that $B$ is contained in at least one of $A$ or $S-A$ is $\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive integers, $n$ is prime, and $m$ and $n$ are relatively prime. Find $m+n+r.$ (The set $S-A$ is the set of all elements of $S$ which are not in $A.$)
MBMT Team Rounds, 2015 F3 E1
Compute $1 - 2 + 3 - 4 + \dots + 2013 - 2014 + 2015$.
1999 AMC 8, 21
The degree measure of angle $A$ is
[asy]
unitsize(12);
draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle);
draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW));
draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW));
draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW));
label(rotate(30)*"$40^\circ$",(2,-8.9),ENE);
label("$100^\circ$",(21/3,-2/3),SE);
label("$110^\circ$",(900/83,-317/83),NNW);
label("$A$",(0,0),NW);[/asy]
$ \text{(A)}\ 20\qquad\text{(B)}\ 30\qquad\text{(C)}\ 35\qquad\text{(D)}\ 40\qquad\text{(E)}\ 45 $
2002 USAMTS Problems, 4
Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$. For example, this gives $f(8)=1$, $f(9)=1$, $f(10)=2$, $f(11)=4$, and $f(12)=5$. Determine the value of $f(10^{100})$.
1989 Tournament Of Towns, (220) 4
A club of $11$ people has a committee. At every meeting of the committee a new committee is formed which differs by $1$ person from its predecessor (either one new member is included or one member is removed). The committee must always have at least three members and , according to the club rules, the committee membership at any stage must differ from its membership at every previous stage. Is it possible that after some time all possible compositions
of the committee will have already occurred?
(S. Fomin , Leningrad)
2008 Bosnia And Herzegovina - Regional Olympiad, 2
IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}\plus{}b^{2}\plus{}c^{2}\equal{}1$ prove the inequality:
\[ \frac{a^{5}\plus{}b^{5}}{ab(a\plus{}b)}\plus{} \frac {b^{5}\plus{}c^{5}}{bc(b\plus{}c)}\plus{}\frac {c^{5}\plus{}a^{5}}{ca(a\plus{}b)}\geq 3(ab\plus{}bc\plus{}ca)\minus{}2.\]
2004 IMO Shortlist, 5
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?
[i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]
2013 NIMO Problems, 8
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$.
[i]Proposed by Lewis Chen[/i]