Found problems: 85335
2014 Contests, 1
Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}$, $|AB|=2$, $|AD|=4$ and $|BC|=|BD|$. If the line through $C$ which is parallel to $AB$ meets $AD$ at $E$, what is $|CE|$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1982 Putnam, B6
Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$
Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have
$ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$
2022 Moldova Team Selection Test, 8
a) Let $n$ $(n \geq 2)$ be an integer. On a line there are $n$ distinct (pairwise distinct) sets of points, such that for every integer $k$ $(1 \leq k \leq n)$ the union of every $k$ sets contains exactly $k+1$ points. Show that there is always a point that belongs to every set.
b) Is the same conclusion true if there is an infinity of distinct sets of points such that for every positive integer $k$ the union of every $k$ sets contains exactly $k+1$ points?
1962 Polish MO Finals, 5
Prove that if $ n $ is a natural number greater than $ 2 $, then $$\sqrt[n + 1]{n+1} < \sqrt[n]{n}.$$
2009 China Girls Math Olympiad, 6
Circle $ \Gamma_{1},$ with radius $ r,$ is internally tangent to circle $ \Gamma_{2}$ at $ S.$ Chord $ AB$ of $ \Gamma_{2}$ is tangent to $ \Gamma_{1}$ at $ C.$ Let $ M$ be the midpoint of arc $ AB$ (not containing $ S$), and let $ N$ be the foot of the perpendicular from $ M$ to line $ AB.$ Prove that $ AC\cdot CB\equal{}2r\cdot MN.$
2006 AIME Problems, 3
Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$.
2014 AMC 12/AHSME, 4
Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?
$ \textbf {(A) } \frac{3}{2} \qquad \textbf {(B) } \frac{5}{3} \qquad \textbf {(C) } \frac{7}{4} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \frac{13}{4}$
2013 Iran Team Selection Test, 9
find all functions $f,g:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $f$ is increasing and also:
$f(f(x)+2g(x)+3f(y))=g(x)+2f(x)+3g(y)$
$g(f(x)+y+g(y))=2x-g(x)+f(y)+y$
2007 Singapore Junior Math Olympiad, 1
Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$. Suppose the area of $\vartriangle DOC$ is $2S/9$. Find the value of $a/b$.
2023 May Olympiad, 1
At Easter Day, $4$ children and their mothers participated in a game in which they had to find hidden chocolate eggs. Augustine found $4$ eggs, Bruno found $6$, Carlos found $9$ and Daniel found $12$. Mrs. Gómez found the same number of eggs as her son, Mrs. Junco found twice as many eggs as her son, Mrs. Messi found three times as many eggs as her son, and Mrs. Núñez found five times as many eggs as her son. At the end of the day, they put all the eggs in boxes, with $18$ eggs in each box, and only one egg was left over. Determine who the mother of each child is.
2010 Sharygin Geometry Olympiad, 21
A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that
\[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]
2009 F = Ma, 17
You are given a standard kilogram mass and a tuning fork that is calibrated in Hz. You are also provided with a complete collection of laboratory equipment, but none of it is calibrated in SI units. You do not know the values of any fundamental constants. Which of the following quantities could you measure in SI units?
(A) The acceleration due to gravity.
(B) The speed of light in a vacuum.
(C) The density of room temperature water.
(D) The spring constant of a given spring.
(E) The air pressure in the room.
2001 AMC 10, 24
In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 12.25 \qquad
\textbf{(C)}\ 12.5 \qquad
\textbf{(D)}\ 12.75 \qquad
\textbf{(E)}\ 13$
1979 Polish MO Finals, 6
A polynomial $w$ of degree $n > 1$ has $n$ distinct zeros $x_1,x_2,...,x_n$. Prove that:
$$\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.$$
Novosibirsk Oral Geo Oly VIII, 2022.5
Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$.
[img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]
Indonesia MO Shortlist - geometry, g1
Given triangle $ABC$, $AL$ bisects angle $\angle BAC$ with $L$ on side $BC$. Lines $LR$ and $LS$ are parallel to $BA$ and $CA$ respectively, $R$ on side $AC$ and$ S$ on side $AB$, respectively. Through point $B$ draw a perpendicular on $AL$, intersecting $LR$ at $M$. If point $D$ is the midpoint of $BC$, prove that that the three points $A, M, D$ lie on a straight line.
2020 China Northern MO, BP1
For all positive real numbers $a,b,c$, prove that
$$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$
1983 Austrian-Polish Competition, 7
Let $P_1,P_2,P_3,P_4$ be four distinct points in the plane. Suppose $\ell_1,\ell_2, … , \ell_6$ are closed segments in that plane with the following property: Every straight line passing through at least one of the points $P_i$ meets the union $\ell_1 \cup \ell_2\cup … \cup\ell_6$ in exactly two points. Prove or disprove that the segments $\ell_i$ necessarily form a hexagon.
2022 Math Prize for Girls Problems, 16
A snail begins a journey starting at the origin of a coordinate plane. The snail moves along line segments of length $\sqrt{10}$ and in any direction such that the horizontal and vertical displacements are both integers. As the snail moves, it leaves a trail tracing out its entire journey. After a while, this trail can form various polygons. What is the smallest possible area of a polygon that could be created by the snail's trail?
1993 Baltic Way, 6
Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ with $2<x<4$ and satisfy: $2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2$ for all $2<x<4$.
Prove that $f(3)=g(3)$.
2024 Baltic Way, 18
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.
1968 AMC 12/AHSME, 8
A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Based on the correct answer, the error thus comitted, to the nearest percent, is:
$\textbf{(A)}\ 100 \qquad
\textbf{(B)}\ 97 \qquad
\textbf{(C)}\ 83 \qquad
\textbf{(D)}\ 17 \qquad
\textbf{(E)}\ 3 $
2000 Harvard-MIT Mathematics Tournament, 1
How many rectangles are there on an $8 \times 8$ checkerboard?
[img]https://cdn.artofproblemsolving.com/attachments/9/e/7719117ae393d81a3e926acb567f850cc1efa9.png[/img]
2024 IFYM, Sozopol, 8
Let $n \geq 2$ be a positive integer. In a mathematics competition, there are $n+1$ students, with one of them being a hacker. The competition is conducted as follows: each receives the same problem with an open-ended answer, has 5 minutes to give their own answer, after which all answers are submitted simultaneously, the correct answer is announced, then they receive a new problem, and so on. The hacker cheats by using spy cameras to see the answers of the other participants. A correct answer gives 1 point, while a wrong answer gives -1 point to everyone except the hacker; for him, it's 0 points because he managed to hack the scoring system. Prove that regardless of the total number of problems, if at some point the hacker is ahead of the second-place contestant by at least $2^{n-2} + 1$ points, then he has a strategy to ensure he will be the sole winner by the end of the competition.
2021 BMT, 24
Suppose that $a, b, c$, and p are positive integers such that $p$ is a prime number and $$a^2 + b^2 + c^2 = ab + bc + ca + 2021p$$.
Compute the least possible value of $\max \,(a, b, c)$.