Found problems: 85335
2012 National Olympiad First Round, 34
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$, what is the least integer $n\geq 2012$?
$ \textbf{(A)}\ 2012 \qquad \textbf{(B)}\ 2013 \qquad \textbf{(C)}\ 2014 \qquad \textbf{(D)}\ 2015 \qquad \textbf{(E)}\ 2016$
2002 JBMO ShortLists, 10
Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that:
$ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$
2007 Korea Junior Math Olympiad, 4
Let $P$ be a point inside $\triangle ABC$. Let the perpendicular bisectors of $PA,PB,PC$ be $\ell_1,\ell_2,\ell_3$. Let $D =\ell_1 \cap \ell_2$ , $E=\ell_2 \cap \ell_3$, $F=\ell_3 \cap \ell_1$. If $A,B,C,D,E,F$ lie on a circle, prove that $C, P,D$ are collinear.
1967 IMO Longlists, 51
A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$
2019 Harvard-MIT Mathematics Tournament, 10
The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$, $a_1 = 4$, and
\[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\]
for $n \ge 0$. Evaluate the sum
\[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\]
1989 AMC 8, 3
Which of the following numbers is the largest?
$\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$
2007 BAMO, 2
The points of the plane are colored in black and white so that whenever three vertices of a parallelogram are the same color, the fourth vertex is that color, too. Prove that all the points of the plane are the same color.
1998 National Olympiad First Round, 29
Let $ ABCD$ be convex quadrilateral with $ \angle C\equal{}\angle D\equal{}90{}^\circ$. The circle $ K$ passing through $ A$ and $ B$ is tangent to $ CD$ at $ C$. Let $ E$ be the intersection of $ K$ and $ \left[AD\right]$. If $ \left|BC\right|\equal{}20$, $ \left|AD\right|\equal{}16$, then $ \left|CE\right|$ is
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6\sqrt{2} \qquad\textbf{(C)}\ 4\sqrt{5} \qquad\textbf{(D)}\ 7\sqrt{2} \qquad\textbf{(E)}\ 10$
2000 Saint Petersburg Mathematical Olympiad, 11.4
Let $P(x)=x^{2000}-x^{1000}+1$. Prove that there don't exist 8002 distinct positive integers $a_1,\dots,a_{8002}$ such that $a_ia_ja_k|P(a_i)P(a_j)P(a_k)$ for all $i\neq j\neq k$.
[I]Proposed by A. Baranov[/i]
2007 Belarusian National Olympiad, 8
Let $(m,n)$ be a pair of positive integers.
(a) Prove that the set of all positive integers can be partitioned into four pairwise disjoint nonempty subsets such that none of them has two numbers with absolute value of their difference equal to either $m$, $n$, or $m+n$.
(b) Find all pairs $(m,n)$ such that the set of all positive integers can not be partitioned into three pairwise disjoint nonempty subsets satisfying the above condition.
2015 Estonia Team Selection Test, 8
Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.
2019 Tournament Of Towns, 5
Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence?
(Alexandr Shapovalov)
2017-IMOC, G1
Given $\vartriangle ABC$. Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$. Let $M, T$ be the midpoints of $BC, PQ$. Show that $MT$ is parallel to the angle bisevtor of $\angle BAC$
[img]http://4.bp.blogspot.com/-MgMtdnPtq1c/XnSHHFl1LDI/AAAAAAAALdY/8g8541DnyGo_Gqd19-7bMBpVRFhbXeYPACK4BGAYYCw/s1600/imoc2017%2Bg1.png[/img]
2004 Denmark MO - Mohr Contest, 1
The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one?
[img]https://1.bp.blogspot.com/-aeqefhbBh5E/XzcBjhgg7sI/AAAAAAAAMXM/B0qSgWDBuqc3ysd-mOitP1LarOtBdJJ3gCLcBGAsYHQ/s0/2004%2BMohr%2Bp1.png[/img]
1999 Czech And Slovak Olympiad IIIA, 1
We are allowed to put several brackets in the expression
$$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$
always in the same places below each other.
(a) Find the smallest possible integer value we can obtain in that way.
(b) Find all possible integer values that can be obtained.
Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.
2017 F = ma, 25
25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is its speed $v_2$ when it passes the perigee?
A) $v_2 = \frac{3}{\sqrt{5}}v_1$
B) $v_2 = \frac{3}{\sqrt{7}}v_1$
C) $v_2 = \frac{2}{\sqrt{3}}v_1$
D) $v_2 = \frac{\sqrt{7}}{\sqrt{3}}v_1$
E) $v_2 = 4v_1$
2020 Online Math Open Problems, 1
Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
[i]Proposed by Ankan Bhattacharya and Brandon Wang[/i]
2022 AMC 10, 20
Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$?
$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$
2022 Girls in Math at Yale, 4
Kara rolls a six-sided die, and if on that first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled?
[i]Proposed by Andrew Wu[/i]
2008 Peru MO (ONEM), 2
Let $a$ and $b$ be real numbers for which the following is true:
$acscx + b cot x \ge 1$, for all $0 <x < \pi$
Find the least value of $a^2 + b$.
1997 Chile National Olympiad, 2
Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$.
Prove that $a \equiv n \equiv 1 \mod 10$ .
2021 Regional Olympiad of Mexico Southeast, 1
Let $A, B$ and $C$ three points on a line $l$, in that order .Let $D$ a point outside $l$ and $\Gamma$ the circumcircle of $\triangle BCD$, the tangents from $A$ to $\Gamma$ touch $\Gamma$ on $S$ and $T$. Let $P$ the intersection of $ST$ and $AC$. Prove that $P$ does not depend of the choice of $D$.
2022 Malaysia IMONST 2, 3
Prove that there is a multiple of $2^{2022}$ that has $2022$ digits, and can be written using digits $1$ and $2$ only.
2023-IMOC, G2
$P$ is a point inside $\triangle ABC$. $AP, BP, CP$ intersects $BC, CA, AB$ at $D, E, F$, respectively. $AD$ meets $(ABC)$ again at $D_1$. $S$ is a point on $(ABC)$. Lines $AS$, $EF$ intersect at $T$, lines $TP, BC$ intersect at $K$, and $KD_1$ meets $(ABC)$ again at $X$. Prove that $S, D, X$ are colinear.
2024 Nigerian MO Round 2, Problem 1
Given a number $\overline{abcd}$, where $a$, $b$, $c$, and $d$, represent the digits of $\overline{abcd}$, find the minimum value of
\[\frac{\overline{abcd}}{a+b+c+d}\]
where $a$, $b$, $c$, and $d$ are distinct
[hide=Answer]$\overline{abcd}=1089$, minimum value of $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$[/hide]