Found problems: 85335
2001 AIME Problems, 9
In triangle $ABC$, $AB=13,$ $BC=15$ and $CA=17.$ Point $D$ is on $\overline{AB},$ $E$ is on $\overline{BC},$ and $F$ is on $\overline{CA}.$ Let $AD=p\cdot AB,$ $BE=q\cdot BC,$ and $CF=r\cdot CA,$ where $p,$ $q,$ and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5.$ The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2005 Purple Comet Problems, 18
The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.
2023 SG Originals, Q3
Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.)
[asy]
unitsize(18);
draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle);
draw((0,1)--(3,1));
draw((2,0)--(2,3));
draw((1,1)--(1,3));
label("A",(0.5,2));
label("B",(1.5,2));
label("C",(2.5,2));
label("D",(1,0.5));
[/asy]
2017 AMC 12/AHSME, 23
The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $24$. What is $f(0)$?
$\textbf{(A) } -2 \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{24}{5} \qquad \textbf{(E) } 8$
1976 AMC 12/AHSME, 22
Given an equilateral triangle with side of length $s$, consider the locus of all points $\mathit{P}$ in the plane of the triangle such that the sum of the squares of the distances from $\mathit{P}$ to the vertices of the triangle is a fixed number $a$. This locus
$\textbf{(A) }\text{is a circle if }a>s^2\qquad$
$\textbf{(B) }\text{contains only three points if }a=2s^2\text{ and is a circle if }a>2s^2\qquad$
$\textbf{(C) }\text{is a circle with positive radius only if }s^2<a<2s^2\qquad$
$\textbf{(D) }\text{contains only a finite number of points for any value of }a\qquad $
$\textbf{(E) }\text{is none of these}$
2004 Finnish National High School Mathematics Competition, 4
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number.
Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?
2022 Math Prize for Girls Olympiad, 1
Let $a$, $b$, $c$ be positive integers with $a \le 10$. Suppose the parabola $y = ax^2 + bx + c$ meets the $x$-axis at two distinct points $A$ and $B$. Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$.
2005 China National Olympiad, 3
As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods.
[img]http://www.mathlinks.ro/Forum/files/china2005_2_214.gif[/img]
Novosibirsk Oral Geo Oly IX, 2022.5
Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.
2022 Turkey Team Selection Test, 1
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
2014 NIMO Summer Contest, 13
Let $\alpha$ and $\beta$ be nonnegative integers. Suppose the number of strictly increasing sequences of integers $a_0,a_1,\dots,a_{2014}$ satisfying $0 \leq a_m \leq 3m$ is $2^\alpha (2\beta + 1)$. Find $\alpha$.
[i]Proposed by Lewis Chen[/i]
2015 Iran Geometry Olympiad, 3
let $ H $ the orthocenter of the triangle $ ABC $
pass two lines $ l_1 $ and $ l_2 $ through $ H $ such that $ l_1 \bot l_2 $
we have $ l_1 \cap BC = D $ and $ l_1 \cap AB = Z $
also $ l_2 \cap BC = E $ and $ l_2 \cap AC = X $ like this picture
pass a line $ d_1$ through $ D $ parallel to $ AC $ and another line $ d_2 $ through $ E $ parallel to $ AB $
let $ d_1 \cap d_2 = Y $
prove $ X $ $ , $ $ Y $ and $ Z $ are on a same line
2009 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle and $A_1$ the foot of the internal bisector of angle $BAC$. Consider $d_A$ the perpendicular line from $A_1$ on $BC$. Define analogously the lines $d_B$ and $d_C$. Prove that lines $d_A, d_B$ and $d_C$ are concurrent if and only if triangle $ABC$ is isosceles.
2005 Gheorghe Vranceanu, 3
$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $
2021 Science ON all problems, 4
$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\
If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\
[i](Vlad Robu)[/i]
2008 Miklós Schweitzer, 1
Let $H \subset P(X)$ be a system of subsets of $X$ and $\kappa > 0$ be a cardinal number such that every $x \in X$ is contained in less than $\kappa$ members of $H$. Prove that there exists an $f \colon X \rightarrow \kappa$ coloring, such that every nonempty $A \in H$ has a “unique” point, that is, an element $x \in A$ such that $f(x) \neq f(y)$ for all $x \neq y \in A$.
(translated by Miklós Maróti)
MathLinks Contest 3rd, 1
Let $a, b, c$ be positive reals. Prove that $$\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.$$
2012 India IMO Training Camp, 1
Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.
2015 Estonia Team Selection Test, 11
Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.
2014 Bundeswettbewerb Mathematik, 2
The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$.
Note: In all the triangles the three vertices do not lie on a straight line.
IV Soros Olympiad 1997 - 98 (Russia), 10.4
Draw on the plane $(p, q)$ all points with coordinates $(p,q)$, for which the equation $\sin^2x+p\sin x+q=0$ has solutions and all its positive solutions form an arithmetic progression.
LMT Guts Rounds, 2019 F
[u]Round 9[/u]
[b]p25.[/b] Find the largest prime factor of $1031301$.
[b]p26.[/b] Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $\angle ABC = 90^o$ , $AB = 5$, $BC = 20$, $CD = 15$. Let $X$, $Y$ be the intersection of the circle with diameter $BC$ and segment $AD$. Find the length of $XY$.
[b]p27.[/b] A string consisting of $1$’s, $2$’s, and $3$’s is said to be a superpermutation of the string $123$ if it contains every permutation of $123$ as a contiguous substring. Find the smallest possible length of such a superpermutation.
[u]Round 10[/u]
[b]p28.[/b] Suppose that we have a function $f (x) = x^3 -3x^2 +3x$, and for all $n \ge 1$, $f^n(x)$ is defined by the function $f$ applied $n$ times to $x$. Find the remainder when $f^5(2019)$ is divided by $100$.
[b]p29.[/b] A function $f : {1,2, . . . ,10} \to {1,2, . . . ,10}$ is said to be happy if it is a bijection and for all $n \in {1,2, . . . ,10}$, $|n - f (n)| \le 1$. Compute the number of happy functions.
[b]p30.[/b] Let $\vartriangle LMN$ have side lengths $LM = 15$, $MN = 14$, and $NL = 13$. Let the angle bisector of $\angle MLN$ meet the circumcircle of $\vartriangle LMN$ at a point $T \ne L$. Determine the area of $\vartriangle LMT$ .
[u]Round 11[/u]
[b]p31.[/b] Find the value of $$\sum_{d|2200} \tau (d),$$ where $\tau (n)$ denotes the number of divisors of $n$, and where $a|b$ means that $\frac{b}{a}$ is a positive integer.
[b]p32.[/b] Let complex numbers $\omega_1,\omega_2, ...,\omega_{2019}$ be the solutions to the equation $x^{2019}-1 = 0$. Evaluate $$\sum^{2019}_{i=1} \frac{1}{1+ \omega_i}.$$
[b]p33.[/b] Let $M$ be a nonnegative real number such that $x^{x^{x^{...}}}$ diverges for all $x >M$, and $x^{x^{x^{...}}}$ converges for all $0 < x \le M$. Find $M$.
[u]Round 12[/u]
[b]p34.[/b] Estimate the number of digits in ${2019 \choose 1009}$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$
[b]p35.[/b] You may submit any integer $E$ from $1$ to $30$. Out of the teams that submit this problem, your score will be $$\frac{E}{2 \, (the\,\, number\,\, of\,\, teams\,\, who\,\, chose\,\, E)}$$
[b]p36.[/b] We call a $m \times n$ domino-tiling a configuration of $2\times 1$ dominoes on an $m\times n$ cell grid such that each domino occupies exactly $2$ cells of the grid and all cells of the grid are covered. How many $8 \times 8$ domino-tilings are there? If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 AIME Problems, 5
A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.
2017 HMNT, 5
[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?
1993 Tournament Of Towns, (380) 2
Vertices $A$, $B$ and $C$ of a triangle are connected with points $A'$ , $B'$ and $C'$ lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments $AA'$, $BB'$ and $CC'$ lie in a straight line?
(Folklore)