Found problems: 85335
Kyiv City MO Juniors 2003+ geometry, 2022.8.3
In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$.
[i](Proposed by Danylo Khilko)[/i]
2023 OMpD, 4
Let $n \geq 0$ be an integer and $f: [0, 1] \rightarrow \mathbb{R}$ an integrable function such that: $$\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1$$ Prove that: $$\int_0^1f(x)^2dx \geq (n+1)^2$$
1997 Slovenia National Olympiad, Problem 4
The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?
2018 CMIMC Team, 8-1/8-2
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$, and let $T = 10X + Y$ for an integer $X$ and a digit $Y$. Suppose that $a$ and $b$ are real numbers satisfying $a+\frac1b=Y$ and $\frac{b}a=X$. Compute $(ab)^4+\frac1{(ab)^4}$.
2012 Argentina Cono Sur TST, 5
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.
2013 JBMO Shortlist, 3
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2019 Teodor Topan, 1
Do exist pairwise distinct matrices $ A,B,C\in \mathcal{M}_2(\mathbb{R}) $ verifying the following properties?
$ \text{(i)} \det A=\det C$
$ \text{(ii)} AB=C,BC=A,CA=B $
$ \text{(iii)} \text{tr} A,\text{tr} B\neq 0 $
[i]Robert Pop[/i]
1999 IberoAmerican, 1
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
2017 District Olympiad, 2
[b]a)[/b] Prove that there exist two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ having the properties:
$ \text{(i)}\quad f\circ g=g\circ f $
$\text{(ii)}\quad f\circ f=g\circ g $
$ \text{(iii)}\quad f(x)\neq g(x), \quad \forall x\in\mathbb{R} $
[b]b)[/b] Show that if there are two functions $ f_1,g_1:\mathbb{R}\longrightarrow\mathbb{R} $ with the properties $ \text{(i)} $ and $ \text{(iii)} $ from above, then $ \left( f_1\circ f_1\right)(x) \neq \left( g_1\circ g_1 \right)(x) , $ for all real numbers $ x. $
1894 Eotvos Mathematical Competition, 3
The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.
2006 USAMO, 6
Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.
2018 Bangladesh Mathematical Olympiad, 5
Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at $G$. The two circles passing through $D$ meet again at $H$. Suppose, $ E, F, G,H $ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof.
1977 AMC 12/AHSME, 13
If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression
$\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad$
$\textbf{(B) }\text{if and only if }a_1=a_2\qquad$
$\textbf{(C) }\text{if and only if }a_1=1\qquad$
$\textbf{(D) }\text{if and only if }a_2=1\qquad $
$\textbf{(E) }\text{if and only if }a_1=a_2=1$
2010 Federal Competition For Advanced Students, Part 1, 3
Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$. A subset $S$ of $M_n$ is called [i]outstanding[/i] if it is non-empty and for every natural number $k\in S$, there exists a $k$-element subset $T_k$ of $S$.
Determine the number $a(n)$ of outstanding subsets of $M_n$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)[/i]
2007 Today's Calculation Of Integral, 205
Evaluate the following definite integral.
\[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]
2016 CentroAmerican, 4
The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is $n$, the player in his/her turn must replace it by an integer $m$ coprime with $n$ and such that $n<m<n^2$. The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.
2021 LMT Fall, 5
In a rectangular prism with volume $24$, the sum of the lengths of its $12$ edges is $60$, and the length of each space diagonal is $\sqrt{109}$. Let the dimensions of the prism be $a\times b\times c$, such that $a>b>c$. Given that $a$ can be written as $\frac{p+\sqrt{q}}{r}$ where $p$, $q$, and $r$ are integers and $q$ is square-free, find $p+q+r$.
2006 IMC, 3
Let $A$ be an $n$x$n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n$x$n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $detB_{i}=b_{i}$ for all $i=1,...,k$.
2016 ASDAN Math Tournament, 15
Circles $\omega_1$ and $\omega_2$ have radii $r_1<r_2$ respectively and intersect at distinct points $X$ and $Y$. The common external tangents intersect at point $Z$. The common tangent closer to $X$ touches $\omega_1$ and $\omega_2$ at $P$ and $Q$ respectively. Line $ZX$ intersects $\omega_1$ and $\omega_2$ again at points $R$ and $S$ and lines $RP$ and $SQ$ intersect again at point $T$. If $XT=8$, $XZ=15$, and $XY=12$, then what is $\tfrac{r_1}{r_2}$?
2005 Oral Moscow Geometry Olympiad, 4
Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular.
(B. Kukushkin)
1994 AMC 12/AHSME, 20
Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$
2017 NIMO Problems, 7
Let the function $f(x) = \left\lfloor x \right\rfloor\{x\}$. Compute the smallest positive integer $n$ such that the graph of $f(f(f(x)))$ on the interval $[0,n]$ is the union of 2017 or more line segments.
[i]Proposed by Ayush Kamat[/i]
2021 AMC 12/AHSME Fall, 10
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40 ^{\circ}, \sin 40 ^{\circ}), (\cos 60 ^{\circ}, \sin 60 ^{\circ}),$ and $(\cos t ^{\circ}, \sin t ^{\circ})$ is isosceles?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 150 \qquad\textbf{(C)}\ 330 \qquad\textbf{(D)}\
360 \qquad\textbf{(E)}\ 380$
2000 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a triangle with $AB = AC$ ¸ $\angle BAC = 100^o$ and $AD, BE$ angle bisectors. Prove that $2AD <BE + EA$
2003 India IMO Training Camp, 8
Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that
(a) triangle $ABC$ is acute,
(b) $a+b+c>r+r_1+r_2+r_3$.