Found problems: 85335
2012 Singapore Junior Math Olympiad, 2
Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?
2012 Hanoi Open Mathematics Competitions, 2
Compare the numbers $P = 2^a,Q = 3, T = 2^b$, where $a=\sqrt2 , b=1+\frac{1}{\sqrt2}$
(A) $P < Q < T$, (B) $T < P < Q$, (C) $P < T < Q$, (D) $T < Q < P$, (E) $ Q < P < T$
1992 Baltic Way, 19
Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.
2023 APMO, 1
Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.
2011 National Olympiad First Round, 19
For which inequality, there exists a line such that the region defined by the inequality and the line intersect in exactly two distinct points?
$\textbf{(A)}\ x^2+y^2\leq 1 \qquad\textbf{(B)}\ |x+y|+|x-y| \leq 1 \qquad\textbf{(C)}\ |x|^3+|y|^3 \leq 1 \\ \textbf{(D)}\ |x|+|y| \leq 1 \qquad\textbf{(E)}\ |x|^{1/2} + |y|^{1/2} \leq 1$
2017 Estonia Team Selection Test, 10
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.
2009 India IMO Training Camp, 2
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
PEN A Problems, 65
Clara computed the product of the first $n$ positive integers and Valerid computed the product of the first $m$ even positive integers, where $m \ge 2$. They got the same answer. Prove that one of them had made a mistake.
1989 Irish Math Olympiad, 2
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is
\[\left( \begin{array}{ccccc}
2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2
\end{array} \right)\]
2023 IFYM, Sozopol, 7
In an acute scalene triangle $ABC$, the incircle $\omega$ touches the sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $P$ be the foot of the perpendicular from $F$ to $DE$. The line $BP$ intersects segment $AC$ at $K$, and the line $AP$ intersects segment $BC$ at $L$. The altitude through vertex $C$ in $\triangle ABC$ intersects the circumcircle of $\triangle CKL$ at a point $Q$. Prove that line $PQ$ passes through the center of $\omega$.
2020 Macedonia Additional BMO TST, 1
Let $P$ and $Q$ be interior points in $\Delta ABC$ such that $PQ$ doesn't contain any vertices of $\Delta ABC$.
Let $A_1$, $B_1$, and $C_1$ be the points of intersection of $BC$, $CA$, and $AB$ with $AQ$, $BQ$, and $CQ$, respectively.
Let $K$, $L$, and $M$ be the intersections of $AP$, $BP$, and $CP$ with $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively.
Prove that $A_1K$, $B_1L$, and $C_1M$ are concurrent.
2007 Estonia Math Open Junior Contests, 8
Call a k-digit positive integer a [i]hyperprime[/i] if all its segments consisting of $ 1, 2, ..., k$ consecutive digits are prime. Find all hyperprimes.
2016 Online Math Open Problems, 27
Let $ABC$ be a triangle with circumradius $2$ and $\angle B-\angle C=15^\circ$. Denote its circumcenter as $O$, orthocenter as $H$, and centroid as $G$. Let the reflection of $H$ over $O$ be $L$, and let lines $AG$ and $AL$ intersect the circumcircle again at $X$ and $Y$, respectively. Define $B_1$ and $C_1$ as the points on the circumcircle of $ABC$ such that $BB_1\parallel AC$ and $CC_1\parallel AB$, and let lines $XY$ and $B_1C_1$ intersect at $Z$. Given that $OZ=2\sqrt 5$, then $AZ^2$ can be expressed in the form $m-\sqrt n$ for positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Michael Ren[/i]
2009 Iran Team Selection Test, 9
In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.
2016 Harvard-MIT Mathematics Tournament, 35
$\textbf{(Maximal Determinant)}$ In a $17 \times 17$ matrix $M$, all entries are $\pm 1$.
The maximum possible value of $\left| \det M \right|$ is $N$. Estimate $N$.
An estimate of $E > 0$ earns $\left\lfloor 20\min(N/E, E/N)^2 \right\rfloor$ points.
2007 Puerto Rico Team Selection Test, 2
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.
2014 ASDAN Math Tournament, 6
Consider a circle of radius $4$ with center $O_1$, a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$, and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$. The centers of the circle are collinear in the order $O_1$, $O_2$, $O_3$. Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$. Compute the length of $AB$.
2017 ITAMO, 3
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make $?$
2010 Dutch BxMO TST, 1
Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.
2021 China Second Round, 3
If $n\ge 4,\ n\in\mathbb{N^*},\ n\mid (2^n-2)$. Prove that $\frac{2^n-2}{n}$ is not a prime number.
2011 Today's Calculation Of Integral, 725
For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$
2023 Bulgarian Autumn Math Competition, 12.2
Given is an acute triangle $ABC$ with incenter $I$ and the incircle touches $BC, CA, AB$ at $D, E, F$. The circle with center $C$ and radius $CE$ meets $EF$ for the second time at $K$. If $X$ is the $C$-excircle touchpoint with $AB$, show that $CX, KD, IF$ concur.
1995 National High School Mathematics League, 6
$O$ is the center of the bottom surface of regular triangular pyramid $P-ABC$. A plane passes $O$ intersects line segment $PC$ at $S$, intersects the extended line of $PA,PB$ at $Q,R$, then $\frac{1}{|PQ|}+\frac{1}{|PR|}+\frac{1}{|PS|}$
$\text{(A)}$ has a maximum value, but no minumum value
$\text{(B)}$ has a minumum value, but no maximum value
$\text{(C)}$ has both minumum value and maximum value (different)
$\text{(D)}$ is a fixed value
2006 Kazakhstan National Olympiad, 6
In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.
2021 Vietnam National Olympiad, 4
For an integer $ n \geq 2 $, let $ s (n) $ be the sum of positive integers not exceeding $ n $ and not relatively prime to $ n $.
a) Prove that $ s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) $, where $ \varphi (n) $ is the number of integers positive cannot exceed $ n $ and are relatively prime to $ n $.
b) Prove that there is no integer $ n \geq 2 $ such that $ s (n) = s (n + 2021) $