This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2008 Germany Team Selection Test, 2
Tags: inequalities , circumcircle , trigonometry , geometry
For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

2014 NIMO Problems, 5
Tags: trigonometry , geometry
In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$? [i]Proposed by Evan Chen[/i]

2017 BMT Spring, 4
Tags: algebra
Find the value of $\frac12+\frac{4}{2^2} +\frac{9}{2^3} +\frac{16}{2^4} + ...$

LMT Speed Rounds, 2010.12
Tags:
Tim is thinking of a positive integer between $2$ and $15,$ inclusive, and Ted is trying to guess the integer. Tim tells Ted how many factors his integer has, and Ted is then able to be certain of what Tim's integer is. What is Tim's integer?

2017-IMOC, A6
Tags: inequalities , algebra
Show that for all positive reals $a,b,c$ with $a+b+c=3$, $$\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.$$

Brazil L2 Finals (OBM) - geometry, 2013.3
Tags: midpoint , projection , geometry , circumcircle , perpendicular
Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

1984 IMO Longlists, 5
Tags: inequalities
For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. If $m \ge 3$, prove that \[\left[\frac{m(m+1)}{2(2m-1)}\right]=\left[\frac{m+1}{4}\right]\]

2007 AMC 10, 14
Tags:
Some boys and girls are having a car wash to raise money for a class trip to China. Initially $ 40 \%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $ 30 \%$ of the group are girls. How many girls were initially in the group? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

1976 Chisinau City MO, 124
Tags: algebra , system of equations
Find $3$ numbers, each of which is equal to the square of the difference of the other two.

2019 BMT Spring, 7
Tags: number theory
How many distinct ordered pairs of integers $(b, m, t)$ satisfy the equation $b^8+m^4+t^2+1 = 2019$?

1999 India National Olympiad, 1
Tags: perimeter , trigonometry , ratio , angle bisector , geometry
Let $ABC$ be an acute-angled triangle in which $D,E,F$ are points on $BC,CA,AB$ respectively such that $AD \perp BC$;$AE = BC$; and $CF$ bisects $\angle C$ internally, Suppose $CF$ meets $AD$ and $DE$ in $M$ and $N$ respectively. If $FM$$= 2$, $MN =1$, $NC=3$, find the perimeter of $\Delta ABC$.

1966 AMC 12/AHSME, 14
Tags: rectangle , ratio , geometry
The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

1971 AMC 12/AHSME, 17
Tags: geometry
A circular disk is divided by $2n$ equally spaced radii($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is $\textbf{(A) }2n+1\qquad\textbf{(B) }2n+2\qquad\textbf{(C) }3n-1\qquad\textbf{(D) }3n\qquad \textbf{(E) }3n+1$

2021 Science ON grade IX, 2
Tags: geometry
Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$. [i] (Călin Pop & Vlad Robu) [/i]

2005 Iran MO (3rd Round), 4
Tags: geometry , trigonometry , trigonometric equation
Suppose in triangle $ABC$ incircle touches the side $BC$ at $P$ and $\angle APB=\alpha$. Prove that : \[\frac1{p-b}+\frac1{p-c}=\frac2{rtg\alpha}\]

2016 India IMO Training Camp, 2
Tags: algebra
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2016 Hong Kong TST, 2
Tags: algebra
Determine all positive integers $n$ for which there exist pairwise distinct positive real numbers $a_1, a_2, \cdots, a_n$ satisfying $\displaystyle \left\{a_i+\frac{(-1)^i}{a_i}\mid 1\leq i \leq n\right\}=\{a_i\mid 1\leq i \leq n\}$

2008 Harvard-MIT Mathematics Tournament, 2
Tags: calculus , integration , function
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?

2023 Dutch Mathematical Olympiad, 3
Tags: number theory , combinatorics
Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the next step: he replaces the number $n$ on the board by $\frac12n$ if $n$ is even and by $n^2 + 3$ if $n$ is odd. For how many choices of starting numbers below $2023$ will Felix never write a number of more than four digits on the board?

1991 Spain Mathematical Olympiad, 5
Tags: number theory , digit sum , binary
For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$.

2020 Estonia Team Selection Test, 1
Tags: geometry
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

2023 Yasinsky Geometry Olympiad, 6
Tags: geometry , circles , circumcircle
In the triangle $ABC$ with sides $AC = b$ and $AB = c$, the extension of the bisector of angle $A$ intersects it's circumcircle at point with $W$. Circle $\omega$ with center at $W$ and radius $WA$ intersects lines $AC$ and $AB$ at points $D$ and $F$, respectively. Calculate the lengths of segments $CD$ and $BF$. (Evgeny Svistunov) [img]https://cdn.artofproblemsolving.com/attachments/7/e/3b340afc4b94649992eb2dccda50ca8f3f7d1d.png[/img]

1985 AIME Problems, 3
Tags: complex number
Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

1966 Miklós Schweitzer, 9
Tags: limit , real analysis
If $ \sum_{m=-\infty}^{+\infty} |a_m| < \infty$, then what can be said about the following expression? \[ \lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.\] [i]P. Turan[/i]

1987 IMO Longlists, 77
Tags: function , inequalities
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]