Found problems: 85335
Durer Math Competition CD Finals - geometry, 2022.C3
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.
2024 Iran MO (3rd Round), 2
Consider the main diagonal and the cells above it in an \( n \times n \) grid. These cells form what we call a tabular triangle of length \( n \). We want to place a real number in each cell of a tabular triangle of length \( n \) such that for each cell, the sum of the numbers in the cells in the same row and the same column (including itself) is zero. For example, the sum of the cells marked with a circle is zero. It is known that the number in the topmost and leftmost cell is $1.$ Find all possible ways to fill the remaining cells.
1962 AMC 12/AHSME, 25
Given square $ ABCD$ with side $ 8$ feet. A circle is drawn through vertices $ A$ and $ D$ and tangent to side $ BC.$ The radius of the circle, in feet, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 4 \sqrt{2} \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 5 \sqrt{2} \qquad
\textbf{(E)}\ 6$
1984 All Soviet Union Mathematical Olympiad, 383
The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?
2023 Stanford Mathematics Tournament, 7
Triangle $ABC$ has $AC = 5$. $D$ and $E$ are on side $BC$ such that $AD$ and $AE$ trisect $\angle BAC$, with $D$ closer to $B$ and $DE =\frac32$, $EC =\frac52$ . From $B$ and $E$, altitudes $BF$ and $EG$ are drawn onto side $AC$. Compute $\frac{CF}{CG}-\frac{AF}{AG}$ .
2010 Contests, 3
Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$
2024 Junior Balkan Team Selection Tests - Romania, P3
Determine all positive integers $a,b,c,d,e,f$ satisfying the following condition: for any two of them, $x{}$ and $y{},$ two of the remaining numbers, $z{}$ and $t{},$ satisfy $x/y=z/t.$
[i]Cristi Săvescu[/i]
2007 AIME Problems, 6
An integer is called [i]parity-monotonic[/i] if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ is $a_{i}$ is even. How many four-digit parity-monotonic integers are there?
2019 Greece National Olympiad, 4
Given a $n\times m$ grid we play the following game . Initially we place $M$ tokens in each of $M$ empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of $M$ in terms of $m,n$ that enables us to win the game.
2003 France Team Selection Test, 2
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
2010 Harvard-MIT Mathematics Tournament, 7
Let $a_1$, $a_2$, and $a_3$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of \[\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.\]
1996 Bundeswettbewerb Mathematik, 4
Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.
2008 Hong Kong TST, 4
Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$.
(a) Show that all such lines $ AB$ are concurrent.
(b) Find the locus of midpoints of all such segments $ AB$.
2013 ISI Entrance Examination, 5
Let $AD$ be a diameter of a circle of radius $r,$ and let $B,C$ be points on the circle such that $AB=BC=\frac r2$ and $A\neq C.$ Find the ratio $\frac{CD}{r}.$
2020 LIMIT Category 1, 12
$q$ is the smallest rational number having the following properties:
(i) $q>\frac{31}{17}$
(ii) when $q$ is written in its reduced form $\frac{a}{b}$, then $b<17$
As in part (ii) above, find $a+b$.
1997 Akdeniz University MO, 2
If $x$ and $y$ are positive reals, prove that
$$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$
2014 Harvard-MIT Mathematics Tournament, 7
Find the maximum possible number of diagonals of equal length in a convex hexagon.
Kvant 2020, M2626
An infinite number of participants gathered for the Olympiad, who were registered under the numbers $1, 2,\ldots$. It turns out that for every $n = 1, 2,\ldots$ a participant with number $n{}$ has at least $n{}$ friends among the remaining participants (note: friendship is mutual). There is a hotel with an infinite number of double rooms. Prove that the participants can be accommodated in double rooms so that there is a couple of friends in each room.
[i]Proposed by V. Bragin, P. Kozhevnikov[/i]
2021 Saint Petersburg Mathematical Olympiad, 4
Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads:
i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites;
ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it.
Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.
1999 All-Russian Olympiad, 3
A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$.
2016 Saudi Arabia Pre-TST, 2.2
Given four numbers $x, y, z, t$, let $(a, b, c, d)$ be a permutation of $(x, y, z, t)$ and set $x_1 =|a- b|$, $y_1 = |b-c|$, $z_1 = |c-d|$, and $t_1 = |d -a|$. From $x_1, y_1, z_1, t_1$, form in the same fashion the numbers $x_2, y_2, z_2, t_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z, t_n = t$ for some $n$. Find all possible values of $(x, y, z, t)$.
2016 Korea Summer Program Practice Test, 8
There are distinct points $A_1, A_2, \dots, A_{2n}$ with no three collinear. Prove that one can relabel the points with the labels $B_1, \dots, B_{2n}$ so that for each $1 \le i < j \le n$ the segments $B_{2i-1} B_{2i}$ and $B_{2j-1} B_{2j}$ do not intersect and the following inequality holds.
\[ B_1 B_2 + B_3 B_4 + \dots + B_{2n-1} B_{2n} \ge \frac{2}{\pi} (A_1 A_2 + A_3 A_4 + \dots + A_{2n-1} A_{2n}) \]
1968 AMC 12/AHSME, 6
Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E$. Let $S$ represent the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC$. If $r=S/S'$, then:
$\textbf{(A)}\ r=1\text{ sometimes, }r>1\text{ sometimes} \qquad\\
\textbf{(B)}\ r=1\text{ sometimes, }r<1\text{ sometimes} \qquad\\
\textbf{(C)}\ 0<r<1\qquad
\textbf{(D)}\ r>1 \qquad
\textbf{(E)}\ r=1 $
2021 Peru IMO TST, P3
Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions:
(i) $f(f(x))=x^2$ for any $x\geq 1$;
(ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$.
1. Prove that $x<f(x)<x^2$ for any $x\geq 1$.
2. Prove that there exists a function $f$ satisfies the above two conditions and the following one:
(iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.
2009 AMC 12/AHSME, 25
The first two terms of a sequence are $ a_1 \equal{} 1$ and $ a_2 \equal{} \frac {1}{\sqrt3}$. For $ n\ge1$,
\[ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} a_{n \plus{} 1}}{1 \minus{} a_na_{n \plus{} 1}}.
\]What is $ |a_{2009}|$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 \minus{} \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 \plus{} \sqrt3$