Found problems: 85335
2010 May Olympiad, 1
Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.
2018 Oral Moscow Geometry Olympiad, 1
Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second.
[img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]
2021-IMOC, A9
For a given positive integer $n,$ find
$$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$
2016 CHMMC (Fall), 12
For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.
1988 Mexico National Olympiad, 2
If $a$ and $b$ are positive integers, prove that $11a+2b$ is a multiple of $19$ if and only if so is $18a+5b$ .
2015 BMT Spring, P1
Suppose $z_0,z_1,\ldots,z_{n-1}$ are complex numbers such that $z_k=e^{2k\pi i/n}$ for $k=0,1,2,\ldots,n-1$. Prove that for any complex number $z$, $\sum_{k=0}^{n-1}|z-z_k|\ge n$.
2019 Greece Junior Math Olympiad, 1
Find all triplets of real numbers $(x,y,z)$ that are solutions to the system of equations
$x^2+y^2+25z^2=6xz+8yz$
$ 3x^2+2y^2+z^2=240$
1991 Hungary-Israel Binational, 4
Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.
2018 ASDAN Math Tournament, 3
The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$?
2011 Indonesia TST, 2
Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$
Kvant 2020, M2625
A connected checkered figure is drawn on a checkered paper. It is known that the figure can be cut both into $2\times 2$ squares and into (possibly rotated) [url=https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tetromino-skew2.svg/1200px-Tetromino-skew2.svg.png]skew-tetrominoes[/url]. Prove that there is a hole in the figure.
[i]Proposed by Y. Markelov and A. Sairanov[/i]
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
2020 Israel National Olympiad, 3
In a convex hexagon $ABCDEF$ the triangles $BDF, ACE$ are equilateral and congruent. Prove that the three lines connecting the midpoints of opposite sides are concurrent.
2007 Today's Calculation Of Integral, 229
Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.
1952 Putnam, A4
The flag of the United Nations consists of a polar map of the world, with the North Pole as its center, extending to approximately $45^\circ$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^\circ$ S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
2021-2022 OMMC, 8
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$. Find $a + b$.
[i]Proposed by Isaac Chen[/i]
2011 Gheorghe Vranceanu, 1
If $ \sqrt{x^2+2y+1} +\sqrt[3]{y^3+3x^2+3x+1} $ is rational, then $ x=y. $
2024 CMI B.Sc. Entrance Exam, 1
(a) Sketch qualitativly the region with maximum area such that it lies in the first quadrant and is bound by $y=x^2-x^3$ and $y=kx$ where $k$ is a constent. The region must not have any other lines closing it. Note: $kx$ lies above $x^2-x^3$
(b) Find an expression for the volume of the solid obtained by spinning this region about the $y$ axis.
2016 India IMO Training Camp, 3
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2002 China Team Selection Test, 1
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.
(1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets.
(2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
OIFMAT II 2012, 2
Find all functions $ f: N \rightarrow N $ such that:
$\bullet$ $ f (m) = 1 \iff m = 1 $;
$\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and
$\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $.
Clarification: $f^n (a) = f (f^{n-1} (a))$
1973 All Soviet Union Mathematical Olympiad, 188
Given $4$ points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?
2019 Lusophon Mathematical Olympiad, 2
Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions:
1. $a + b + c = n$
2. $ax^2 + bx + c = 0$ has rational roots.
1997 French Mathematical Olympiad, Problem 3
Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$.
2013 Princeton University Math Competition, 12
Let $D$ be a point on the side $BC$ of $\triangle ABC$. If $AB=8$, $AC=7$, $BD=2$, and $CD=1$, find $AD$.