Found problems: 85335
2018 MIG, 14
How many integers between $80$ and $100$ are prime?
$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$
2019 District Olympiad, 2
Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and $B'$ respectively on the diagonal $BD'$.
a) Prove that $BM + BN + BP = BD'$.
b) Prove that $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.
2016 Hanoi Open Mathematics Competitions, 5
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to
(A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.
2014 Rioplatense Mathematical Olympiad, Level 3, 6
Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$.
Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .
1972 USAMO, 2
A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.
2010 IFYM, Sozopol, 5
Each vertex of a right $n$-gon $(n\geq 3)$ is colored in yellow, blue or red. On each turn are chosen two adjacent vertices in different color and then are recolored in the third. For which $n$ can we get from an arbitrary coloring of the $n$-gon a monochromatic one (in one color)?
1992 Poland - First Round, 7
Given are the points $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$ in the space. Let $P_{ij} (i,j \in 0,1,2,3)$ be the point determined by the equality: $\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}$. Find the volume of the smallest convex polyhedron which contains all the points $P_{ij}$.
2020 Turkey Team Selection Test, 9
For $a,n$ positive integers we show number of different integer 10-tuples $ (x_1,x_2,...,x_{10})$ on $ (mod n)$ satistfying $x_1x_2...x_{10}=a (mod n)$ with $f(a,n)$. Let $a,b$ given positive integers ,
a) Prove that there exist a positive integer $c$ such that for all $n\in \mathbb{Z^+}$ $$\frac {f(a,cn)}{f(b,cn)}$$is constant
b) Find all $(a,b)$ pairs such that minumum possible value of $c$ is 27 where $c$ satisfying condition in $(a)$
2012 Online Math Open Problems, 9
At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?
[i]Author: Ray Li[/i]
2005 AMC 12/AHSME, 8
Let $ A$, $ M$, and $ C$ be digits with
\[ (100A \plus{} 10M \plus{} C )(A \plus{} M \plus{} C ) \equal{} 2005.
\]What is $ A$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2010 China Northern MO, 7
Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$
The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.
2025 China Team Selection Test, 19
Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer \( n \), the following holds:
\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]
2016 India IMO Training Camp, 2
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2025 Turkey Team Selection Test, 6
Let $ABC$ be a scalene triangle with incenter $I$ and incircle $\omega$. Let the tangency points of $\omega$ to $BC,AC\text{ and } AB$ be $D,E,F$ respectively. Let the line $EF$ intersect the circumcircle of $ABC$ at the points $G, H$. Assume that $E$ lies between the points $F$ and $G$. Let $\Gamma$ be a circle that passes through $G$ and $H$ and that is tangent to $\omega$ at the point $M$ which lies on different semi-planes with $D$ with respect to the line $EF$. Let $\Gamma$ intersect $BC$ at points $K$ and $L$ and let the second intersection point of the circumcircle of $ABC$ and the circumcircle of $AKL$ be $N$. Prove that the intersection point of $NM$ and $AI$ lies on the circumcircle of $ABC$ if and only if the intersection point of $HB$ and $GC$ lies on $\Gamma$.
2020 Iranian Geometry Olympiad, 5
We say two vertices of a simple polygon are [i]visible[/i] from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers $n$ such that there exists a simple polygon with $n$ vertices in which every vertex is visible from exactly $4$ other vertices.
(A simple polygon is a polygon without hole that does not intersect itself.)
[i]Proposed by Morteza Saghafian[/i]
2019-IMOC, N2
Find all pairs of positive integers $(m, n)$ such that
$$m^n * n^m = m^m + n^n$$
2015 Turkey Team Selection Test, 3
Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that
\[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]
1969 Swedish Mathematical Competition, 6
Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?
2018 Iran Team Selection Test, 3
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.
[i]Proposed by Iman Maghsoudi[/i]
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 $ABC$.