Found problems: 85335
2021 Germany Team Selection Test, 3
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
$\emph{Slovakia}$
2016 Korea Winter Program Practice Test, 4
Let $a_1, a_2, \cdots a_{100}$ be a permutation of $1,2,\cdots 100$.
Define $l(k)$ as the maximum $m$ such that there exists $i_1, i_2 \cdots i_m$ such that $a_{i_1} > a_{i_2} > \cdots > a_{i_m}$ or $a_{i_1} < a_{i_2} < \cdots < a_{i_m}$, where $i_1=k$ and $i_1<i_2< \cdots <i_m$
Find the minimum possible value for $\sum_{i=1}^{100} l(i)$.
KoMaL A Problems 2021/2022, A. 814
We are given $666$ points on the plane such that they cannot be covered by $10$ lines. Show that we can choose $66$ out of these points such that they can not be covered by $10$ lines.
2017 USAMTS Problems, 2
Let $b$ be a positive integer. Grogg writes down a sequence whose first term is $1$. Each term after that is the total number of digits in all the previous terms of the sequence when written in base $b$. For example, if $b = 3$, the sequence starts $1, 1, 2, 3, 5, 7, 9, 12, \dots$. If $b = 2521$, what is the first positive power of $b$ that does not appear in the sequence?
2012 Tournament of Towns, 6
(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines.
(b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.
2019 Taiwan APMO Preliminary Test, P1
In $\triangle ABC$, $\angle B=90^\circ$, segment $AB>BC$. Now we have a $\triangle A_iBC(i=1,2,...,n)$ which is similiar to $\triangle ABC$ (the vertexs of them might not correspond). Find the maximum value of $n+2018$.
1975 Bundeswettbewerb Mathematik, 1
Let $a, b, c, d$ be distinct positive real numbers. Prove that if one of the numbers $c, d$ lies between $a$ and $b$, or one of $a, b$ lies between $c$ and $d$, then
$$\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}$$
and that otherwise, one can choose $a, b, c, d$ so that this inequality is false.
2013 Junior Balkan Team Selection Tests - Moldova, 7
The points $M$ and $N$ are located respectively on the diagonal $(AC)$ and the side $(BC)$ of the square $ABCD$ such that $MN = MD$. Determine the measure of the angle $MDN$.
1978 IMO Shortlist, 2
Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.
Durer Math Competition CD 1st Round - geometry, 2016.C+3
Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum?
2006 AMC 10, 10
For how many real values of $ x$ is $ \sqrt {120 \minus{} \sqrt {x}}$ an integer?
$ \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11$
2014 Contests, 1
Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.
2017 Tuymaada Olympiad, 7
An equilateral triangle with side $20$ is divided by there series of parallel lines into $400$ equilateral triangles with side $1$. What maximum number of these small triangles can be crossed (internally) by one line?
Tuymaada 2017 Q7 Juniors
1963 AMC 12/AHSME, 25
Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of triangle $CEF$ is $200$ square inches. Then the number of inches in $BE$ is:
[asy]
size(6cm);
pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7);
draw(A--B--C--D--cycle);
draw(F--C--E--B);
label("$A$", A, SW);
label("$B$", B, S);
label("$C$", C, N);
label("$D$", D, NW);
label("$E$", E, SE);
label("$F$", F, W);
//Credit to MSTang for the asymptote
[/asy]
$\textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 20$
2020 Iran Team Selection Test, 2
Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$.
[i]Proposed by Alireza Dadgarnia[/i]
2006 China Team Selection Test, 2
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
2012 Serbia JBMO TST, 3
Let $a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci}, \overline{dea}, \overline{ifd}, \overline{jgf}, \overline{bfeg},\ldots$ be an increasing arithmetic progression. Find the $16$th term of this sequence.
2009 IMO Shortlist, 4
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
2024 Kyiv City MO Round 1, Problem 3
The circle $\gamma$ passing through the vertex $A$ of triangle $ABC$ intersects its sides $AB$ and $AC$ for the second time at points $X$ and $Y$, respectively. Also, the circle $\gamma$ intersects side $BC$ at points $D$ and $E$ so that $AD = AE$. Prove that the points $B, X, Y, C$ lie on the same circle.
[i]Proposed by Mykhailo Shtandenko[/i]
1985 All Soviet Union Mathematical Olympiad, 414
Solve the equation ("$2$" encounters $1985$ times):
$$\dfrac{x}{2+ \dfrac{x}{2+\dfrac{x}{2+... \dfrac{x}{2+\sqrt {1+x}}}}}=1$$
1996 Singapore MO Open, 3
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.
1996 Singapore MO Open, 4
Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer
2023 Pan-American Girls’ Mathematical Olympiad, 6
Let $n \geq 2$ be an integer. Lucia chooses $n$ real numbers $x_1,x_2,\ldots,x_n$ such that $\left| x_i-x_j \right|\geq 1$ for all $i\neq j$. Then, in each cell of an $n \times n$ grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the $n^2$ numbers that Lucia calculated can take.
2023 Stanford Mathematics Tournament, 3
Let $f(x)=x^3-6x^2+\tfrac{25}{2}x-7$. There is an interval $[a,b]$ such that for any real number $x$, the sequence $x,f(x),f(f(x)),\dots$ is bounded (i.e., has a lower and upper bound) if and only if $x\in[a,b]$. Compute $(a-b)^2$.
2023 Vietnam Team Selection Test, 4
Given are two coprime positive integers $a, b$ with $b$ odd and $a>2$. The sequence $(x_n)$ is defined by $x_0=2, x_1=a$ and $x_{n+2}=ax_{n+1}+bx_n$ for $n \geq 1$. Prove that:
$a)$ If $a$ is even then there do not exist positive integers $m, n, p$ such that $\frac{x_m} {x_nx_p}$ is a positive integer.
$b)$ If $a$ is odd then there do not exist positive integers $m, n, p$ such that $mnp$ is even and $\frac{x_m} {x_nx_p}$ is a perfect square.