Found problems: 85335
1996 AMC 12/AHSME, 29
If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have?
$\text{(A)}\ 32 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 36\qquad \text{(E)}\ 38$
2010 All-Russian Olympiad, 3
Quadrilateral $ABCD$ is inscribed into circle $\omega$, $AC$ intersect $BD$ in point $K$. Points $M_1$, $M_2$, $M_3$, $M_4$-midpoints of arcs $AB$, $BC$, $CD$, and $DA$ respectively. Points $I_1$, $I_2$, $I_3$, $I_4$-incenters of triangles $ABK$, $BCK$, $CDK$, and $DAK$ respectively. Prove that lines $M_1I_1$, $M_2I_2$, $M_3I_3$, and $M_4I_4$ all intersect in one point.
2015 Indonesia MO Shortlist, G7
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.
2004 Iran MO (3rd Round), 11
assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle at A" in the point $X_a$; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.
2011 Kazakhstan National Olympiad, 6
Determine all pairs of positive real numbers $(a, b)$ for which there exists a function $ f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}} $ satisfying for all positive real numbers $x$ the equation
$ f(f(x))=af(x)- bx $
2011 Polish MO Finals, 3
Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations
\[\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.\]
2000 All-Russian Olympiad Regional Round, 11.1
Prove that it is possible to choose different real numbers $a_1, a_2, . . . , a_{10}$ that the equation $$(x - a_1)(x -a_2).... (x -a_{10}) = (x + a_1)(x + a_2) ...(x + a_{10})$$ will have exactly $5$ different real roots.
2016 ELMO Problems, 5
Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$, Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of $n$, what is the minimum number of colors Elmo could have used?
[i]Michael Ren[/i]
2018 Romania National Olympiad, 1
Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$
2003 IMO Shortlist, 3
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
2012 Serbia JBMO TST, 2
Show that the equation $x^2+y^2+z^2-xy-yz-zx=3$ has an infinity solutions over nonnegative integers.
2019 Oral Moscow Geometry Olympiad, 5
Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.
2013 AIME Problems, 2
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5$,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5$.
2023 IFYM, Sozopol, 7
In the country of Drilandia, which has at least three cities, there are bidirectional roads connecting some pairs of cities such that any city can be reached from any other. Two cities are called [i]close[/i] if one can reach the other by using at most two intermediary cities. The mayor, Drilago, fortified the road system by building a direct road between each pair of close cities that were not already connected. Prove that after the expansion, there exists a journey that starts and ends at the same city, where each city except the first is visited exactly once, and the first city is visited twice (once at the beginning and once at the end).
2019 India PRMO, 4
An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?
1997 Cono Sur Olympiad, 5
Let $n$ be a natural number $n>3$.
Show that in the multiples of $9$ less than $10^n$, exist more numbers with the sum of your digits equal to $9(n - 2)$ than numbers with the sum of your digits equal to $9(n - 1)$.
PEN Q Problems, 12
Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
2007 Alexandru Myller, 4
At a math contest which has $ 5 $ problems, each candidate has solved $ 3 $ problems. Among these candidates, for any group of $ 5 $ candidates that we might choose, we see that there is a problem which all members of the group had solved it. Prove that there is a problem solved by all candidates.
1986 ITAMO, 2
Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$
.
2000 Mediterranean Mathematics Olympiad, 2
Suppose that in the exterior of a convex quadrilateral $ABCD$ equilateral triangles $XAB,YBC,ZCD,WDA$ with centroids $S_1,S_2,S_3,S_4$ respectively are constructed. Prove that $S_1S_3\perp S_2S_4$ if and only if $AC=BD$.
2021 Estonia Team Selection Test, 3
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
2020 LMT Fall, 27
A list consists of all positive integers from $1$ to $2020$, inclusive, with each integer appearing exactly once. Define a move as the process of choosing four numbers from the current list and replacing them with the numbers $1,2,3,4$. If the expected number of moves before the list contains exactly two $4$'s can be expressed as $\frac{a}{b}$ for relatively prime positive integers, evaluate $a+b$.
[i]Proposed by Richard Chen and Taiki Aiba[/i]
2018 International Zhautykov Olympiad, 2
Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$.Let $BL\cap AK=P$.If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$
1985 IMO Longlists, 94
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.
1997 Putnam, 4
Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that :
\[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \]
Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$
Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )