Found problems: 85335
2011 Irish Math Olympiad, 1
Prove that $$\frac{2}{3}+\frac{4}{5}+\dots +\frac{2010}{2011}$$ is not an integer.
2014 Saudi Arabia Pre-TST, 1.2
Let $D$ be the midpoint of side $BC$ of triangle $ABC$ and $E$ the midpoint of median $AD$. Line $BE$ intersects side $CA$ at $F$. Prove that the area of quadrilateral $CDEF$ is $\frac{5}{12}$ the area of triangle $ABC$.
CIME II 2018, 6
Define $f(x)=-\frac{2x}{4x+3}$ and $g(x)=\frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x)=g(f(h^n(x)))$, where $h^1(x)=g(f(x))$. If the value of $\sum_{k=1}^{100} (-1)^k\cdot h^{100}(k)$ can be written in the form $ab^c$, for some integers $a,b,c$ where $c$ is as maximal as possible and $b\ne 1$, find $a+b+c$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2023 Princeton University Math Competition, A1 / B3
Find the integer $x$ for which $135^3+138^3=x^3-1.$
2021 Harvard-MIT Mathematics Tournament., 10
Let $S$ be a set of positive integers satisfying the following two conditions:
• For each positive integer $n$, at least one of $n, 2n, \dots, 100n$ is in $S$.
• If $a_1, a_2, b_1, b_2$ are positive integers such that $\gcd(a_1a_2, b_1b_2) = 1$ and $a_1b_1, a_2b_2 \in S,$ then
$a_2b_1, a_1b_2 \in S.$
Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\lfloor 10^5r\rfloor$.
Note: $S$ has natural density $r$ if $\tfrac{1}{n}|S \cap {1, \dots, n}|$ approaches $r$ as $n$ approaches $\infty$.
2014 Contests, 3
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
LMT Team Rounds 2021+, A13
In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students $A$, $B$, and $C$, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter.
[i]Proposed by Kevin Zhao[/i]
1970 IMO Longlists, 21
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
1958 Miklós Schweitzer, 1
[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]
2000 Moldova National Olympiad, Problem 5
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
2012 District Olympiad, 4
For all odd natural numbers $ n, $ prove that
$$ \left|\sum_{j=0}^{n-1} (a+ib)^j\right|\in\mathbb{Q} , $$
where $ a,b\in\mathbb{Q} $ are two numbers such that $ 1=a^2+b^2. $
2020 Stanford Mathematics Tournament, 9
Let $ABC$ be a right triangle with hypotenuse $AC$. Let $G$ be the centroid of this triangle and suppose that we have $AG^2 + BG^2 + CG^2 = 156$. Find $AC^2$.
Kvant 2019, M2585
Let $a_1,...,a_n$ be $n$ real numbers. If for each odd positive integer $k\leqslant n$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$, then for each odd positive integer $k$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$.
[i]Proposed by M. Didin[/i]
2021 Simon Marais Mathematical Competition, A3
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
[i]Here $\det A$ denotes the determinant of the matrix $A$.[/i]
2013 China Team Selection Test, 2
The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively.
Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$
2005 Bundeswettbewerb Mathematik, 1
In the centre of a $2005 \times 2005$ chessboard lies a dice that is to be moved across the board in a sequence of moves.
One move consists of the following three steps:
- The dice has to be turned with an arbitrary side on top,
- then it has to be moved by the shown number of points to the right or left
- and finally moved by the concealed number of points upwards or downwards.
The attained square is the starting square for the next move.
Which squares of the chessboard can be reached in a finite sequence of such moves?
2007 Princeton University Math Competition, 3
Find all values of $b$ such that the difference between the maximum and minimum values of $f(x) = x^2-2bx-1$ on the interval $[0, 1]$ is $1$.
2011 Saudi Arabia BMO TST, 1
Let $n$ be a positive integer. Find all real numbers $x_1,x_2 ,..., x_n$ such that $$\prod_{k=1}^{n}(x_k^2+ (k + 2)x_k + k^2 + k + 1) =\left(\frac{3}{4}\right)^n (n!)^2$$
2018 PUMaC Number Theory B, 2
Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.
MathLinks Contest 6th, 4.3
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
$$\sqrt{\frac{a+b}{b+1}}+\sqrt{\frac{b+c}{c+1}}+\sqrt{\frac{c+a}{a+1}} \ge 3$$
2021 BMT, 4
An equilateral polygon has unit side length and alternating interior angle measures of $15^o$ and $300^o$. Compute the area of this polygon.
1979 Romania Team Selection Tests, 5.
a) Are there rectangles $1\times \dfrac12$ rectangles lying strictly inside the interior of a unit square?
b) Find the minimum number of equilateral triangles of unit side which can cover completely a unit square.
[i]Laurențiu Panaitopol[/i]
2003 China Team Selection Test, 1
There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.
2010 Sharygin Geometry Olympiad, 5
Let $AH$, $BL$ and $CM$ be an altitude, a bisectrix and a median in triangle $ABC$. It is known that lines $AH$ and $BL$ are an altitude and a bisectrix of triangle $HLM$. Prove that line $CM$ is a median of this triangle.
2019 Jozsef Wildt International Math Competition, W. 2
If $0<a\leq c\leq b$ then $$\frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36b^{10}}\leq \frac{(b^{25}-a^{25})(b^{25}-c^{25})}{25}\leq \frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36(ac)^{10}}$$