Found problems: 85335
2024 Caucasus Mathematical Olympiad, 8
There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$.
(a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles.
(b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.
2009 Federal Competition For Advanced Students, P1, 2
For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$
$(n-2k)$...$(r)$, where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$
Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square.
2008 Bosnia And Herzegovina - Regional Olympiad, 2
If $ a$, $ b$ and $ c$ are positive reals prove inequality:
\[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]
2011 Canadian Students Math Olympiad, 1
In triangle $ABC$, $\angle{BAC}=60^\circ$ and the incircle of $ABC$ touches $AB$ and $AC$ at $P$ and $Q$, respectively. Lines $PC$ and $QB$ intersect at $G$. Let $R$ be the circumradius of $BGC$. Find the minimum value of $R/BC$.
[i]Author: Alex Song[/i]
2013 VJIMC, Problem 2
Let $A=(a_{ij})$ and $B=(b_{ij})$ be two real $10\times10$ matrices such that $a_{ij}=b_{ij}+1$ for all $i,j$ and $A^3=0$. Prove that $\det B=0$.
2015 Tournament of Towns, 6
Basil has a melon in a shape of a ball, $20$ in diameter. Using a long knife, Basil makes three mutually perpendicular cuts. Each cut carves a circular segment in a plane of the cut, $h$ deep ($h$ is a height of the segment). Does it necessarily follow that the melon breaks into two or more pieces if
(a) $h = 17$ ? [i](6 points)[/i]
(b) $h = 18$ ? [i](6 points)[/i]
2006 Tournament of Towns, 3
Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $100 < xa < 1000$ given that inequality $10 < xa < 100$ has exactly $5$ integer solutions. Consider all possible cases.
[i](4 points)[/i]
2000 Junior Balkan Team Selection Tests - Moldova, 2
The number $665$ is represented as a sum of $18$ natural numbers nenule $a_1, a_2, ..., a_{18}$.
Determine the smallest possible value of the smallest common multiple of the numbers $a_1, a_2, ..., a_{18}$.
1997 May Olympiad, 2
In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$
2007 ITest, 14
Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)$ equal to $12$?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$
$\textbf{(J) }9\hspace{14.2em}\textbf{(K) }10\hspace{13.5em}\textbf{(L) }11$
$\textbf{(M) }12\hspace{13.3em}\textbf{(N) }13$
2000 Putnam, 4
Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.
2011 Dutch BxMO TST, 2
In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.
1968 AMC 12/AHSME, 21
If $S=1!+2!+3!+ \cdots +99!$, then the units' digit in the value of $S$ is:
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 0$
1986 IMO Longlists, 16
Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties:
\[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\]
\[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]
2007 Harvard-MIT Mathematics Tournament, 5
Compute the largest positive integer such that $\dfrac{2007!}{2007^n}$ is an integer.
2007 National Olympiad First Round, 33
The tangent lines from the point $A$ to the circle $C$ touches the circle at $M$ and $N$. Let $P$ a point on $[AN]$. Let $MP$ meet $C$ at $Q$. Let $MN$ meet the line through $P$ and parallel to $MA$ at $R$. If $|MA|=2$, $|MN|=\sqrt 3$, and $QR \parallel AN$, what is $|PN|$?
$
\textbf{(A)}\ \dfrac 32
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \dfrac {\sqrt 3} 2
\qquad\textbf{(D)}\ \sqrt 2
\qquad\textbf{(E)}\ \sqrt 3
$
2023 Thailand TST, 2
Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.
2009 National Olympiad First Round, 36
There are one-way flights between $100$ cities of a country. It is possible to fly starting from the capital city and visiting all other $99$ cities and returning again to the capital city. Let $ N$ be the smallest number of flights inorder to form such a flight combination. Among all flight combinations (satisfying previous condtions), $ N$ can be at most ?
$\textbf{(A)}\ 1850 \qquad\textbf{(B)}\ 2100 \qquad\textbf{(C)}\ 2550 \qquad\textbf{(D)}\ 3060 \qquad\textbf{(E)}\ \text{None}$
2001 Austrian-Polish Competition, 4
Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$, then the following inequality holds \[S \leq \frac{1}{2}(ac+bd).\] For which quadrangles does the inequality become equality?
2002 AMC 10, 14
The vertex $E$ of a square $EFGH$ is at the center of square $ABCD$. The length of a side of $ABCD$ is $1$ and the length of a side of $EFGH$ is $2$. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID=60^\circ$, the area of quadrilateral $EIDJ$ is
$\textbf{(A) }\dfrac14\qquad\textbf{(B) }\dfrac{\sqrt3}6\qquad\textbf{(C) }\dfrac13\qquad\textbf{(D) }\dfrac{\sqrt2}4\qquad\textbf{(E) }\dfrac{\sqrt3}2$
2022/2023 Tournament of Towns, P2
A big circle is inscribed in a rhombus, each of two smaller circles touches two sides of the rhombus and the big circle as shown in the figure on the right. Prove that the four dashed lines spanning the points where the circles touch the rhombus as shown in the figure make up a square.
1993 AMC 12/AHSME, 3
$\frac{15^{30}}{45^{15}}=$
$ \textbf{(A)}\ \left(\frac{1}{3}\right)^{15} \qquad\textbf{(B)}\ \left(\frac{1}{3}\right)^2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 3^{15} \qquad\textbf{(E)}\ 5^{15}$
2019 Online Math Open Problems, 26
Let $p = 491$ be prime. Let $S$ be the set of ordered $k$-tuples of nonnegative integers that are less than $p$. We say that a function $f\colon S \to S$ is \emph{$k$-murine} if, for all $u,v\in S$, $\langle f(u), f(v)\rangle \equiv \langle u,v\rangle \pmod p$, where $\langle(a_1,\dots ,a_k) , (b_1, \dots , b_k)\rangle = a_1b_1+ \dots +a_kb_k$ for any $(a_1, \dots a_k), (b_1, \dots b_k) \in S$.
Let $m(k)$ be the number of $k$-murine functions. Compute the remainder when $m(1) + m(2) + m(3) + \cdots + m(p)$ is divided by $488$.
[i]Proposed by Brandon Wang[/i]
2006 Junior Balkan Team Selection Tests - Moldova, 3
The convex polygon $A_{1}A_{2}\ldots A_{2006}$ has opposite sides parallel $(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)$.
Prove that the diagonals $A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}$ are concurrent if and only if opposite sides are equal.
2014-2015 SDML (Middle School), 8
If the five-digit number $3AB7C$ is divisible by $4$ and $9$ and $A<B<C$, what is $A+B+C$?
$\text{(A) }3\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }17\qquad\text{(E) }26$