Found problems: 85335
Kyiv City MO Seniors 2003+ geometry, 2006.11.3
Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$, and $W$ be the midpoint of the arc $BC$ of the circle $\omega$, which does not contain the point $A$, and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$. Find the angle $\angle BAC$, if $WO = WH$.
(O. Clurman)
2001 National Olympiad First Round, 17
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$, what is the largest possible area of the triangle?
$
\textbf{(A)}\ 20
\qquad\textbf{(B)}\ 25
\qquad\textbf{(C)}\ 30
\qquad\textbf{(D)}\ 40
\qquad\textbf{(E)}\ 50
$
2020 Greece Junior Math Olympiad, 3
Find all positive integers $x$, for which the equation
$$a+b+c=xabc$$ has solution in positive integers.
Solve the equation for these values of $x$
2021 JHMT HS, 8
For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
1949-56 Chisinau City MO, 38
Which is more $\log_3 7$ or $\log_{\frac{1}{3}} \frac{1}{7}$ ?
2019 ELMO Shortlist, G6
Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$.
a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$.
b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$.
[i]Proposed by Vincent Huang[/i]
2014 Contests, 1
Say that a convex quadrilateral is [i]tasty[/i] if its two diagonals divide the quadrilateral into four nonoverlapping similar triangles. Find all tasty convex quadrilaterals. Justify your answer.
2012 Hanoi Open Mathematics Competitions, 12
In an isosceles triangle ABC with the base AB
given a point M $\in$ BC: Let O be the center of its circumscribed
circle and S be the center of the inscribed circle in ABC and
SM // AC: Prove that OM perpendicular BS.
1952 AMC 12/AHSME, 5
The points $ (6,12)$ and $ (0, \minus{} 6)$ are connected by a straight line. Another point on this line is:
$ \textbf{(A)}\ (3,3) \qquad\textbf{(B)}\ (2,1) \qquad\textbf{(C)}\ (7,16) \qquad\textbf{(D)}\ ( \minus{} 1, \minus{} 4) \qquad\textbf{(E)}\ ( \minus{} 3, \minus{} 8)$
2023 Ukraine National Mathematical Olympiad, 10.3
Let $I$ be the incenter of the triangle $ABC$, and $P$ be any point on the arc $BAC$ of its circumcircle. Points $K$ and $L$ are chosen on the tangent to the circumcircle $\omega$ of triangle $API$ at point $I$, so that $BK = KI$ and $CL = LI$. Show that the circumcircle of triangle $PKL$ is tangent to $\omega$.
[i]Proposed by Mykhailo Shtandenko[/i]
2007 Regional Competition For Advanced Students, 1
Let $ 0<x_0,x_1, \dots , x_{669}<1$ be pairwise distinct real numbers. Show that there exists a pair $ (x_i,x_j)$ with
$ 0<x_ix_j(x_j\minus{}x_i)<\frac{1}{2007}$
2017 Cono Sur Olympiad, 5
Let $a$, $b$ and $c$ positive integers. Three sequences are defined as follows:
[list]
[*] $a_1=a$, $b_1=b$, $c_1=c$[/*]
[*] $a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$, $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$, $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$[/*]
[/list]
[list = a]
[*]Prove that for any $a$, $b$, $c$, there exists a positive integer $N$ such that $a_N=b_N=c_N$.[/*]
[*]Find the smallest $N$ such that $a_N=b_N=c_N$ for some choice of $a$, $b$, $c$ such that $a \ge 2$ y $b+c=2a-1$.[/*]
[/list]
2017 AMC 8, 18
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$.
[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0));
label("$B$", (0, 0), SW);
label("$A$", (12, 0), ESE);
label("$C$", (2.4, 3.6), SE);
label("$D$", (0, 5), N);[/asy]
What is the area of quadrilateral $ABCD$?
$\textbf{(A) }12\qquad\textbf{(B) }24\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }36$
2006 AIME Problems, 7
Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.
2022 Math Prize for Girls Problems, 2
Let $b$ and $c$ be random integers from the set $\{1, 2, \ldots, 100\}$, chosen uniformly and independently. What is the probability that the roots of the quadratic $x^2 + bx + c$ are real?
1966 AMC 12/AHSME, 1
Given that the ratio of $3x-4$ to $y+15$ is constant, and $y=3$ when $x=2$, then, when $y=12$, $x$ equals:
$\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 8$
2012 Purple Comet Problems, 13
Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.
2021 CMIMC Integration Bee, 4
$$\int_1^2\left(x^5+6x^4+14x^3+16x^2+9x+2\right)dx$$
[i]Proposed by Connor Gordon[/i]
1992 Bulgaria National Olympiad, Problem 6
There are given one black box and $n$ white boxes ($n$ is a random natural number). White boxes are numbered with the numbers $1,2,\ldots,n$. In them are put $n$ balls. It is allowed the following rearrangement of the balls: if in the box with number $k$ there are exactly $k$ balls, that box is made empty - one of the balls is put in the black box and the other $k-1$ balls are put in the boxes with numbers: $1,2,\ldots,k-1$. [i](Ivan Tonov)[/i]
2021 AMC 10 Spring, 9
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$?
$\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$
1996 AMC 8, 12
What number should be removed from the list
\[1,2,3,4,5,6,7,8,9,10,11\]
so that the average of the remaining numbers is $6.1$?
$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$
1950 AMC 12/AHSME, 44
The graph of $ y\equal{}\log x$
$\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\
\textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\
\textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\
\textbf{(D)}\ \text{Cuts neither axis} \qquad\\
\textbf{(E)}\ \text{Cuts all circles whose center is at the origin}$
2021-2022 OMMC, 19
$N$ people have a series of calls. Each call is between two people, and is started by exactly one of them. Each person starts at most $10$ calls. Two people can call at most once. In any group of $3$ people, there are at least two people who have a call. Find the maximum possible value of $N$.
[i]Proposed by Serena Xu[/i]
1987 China National Olympiad, 4
Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$. Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied:
i) the five points are covered by the three equilateral triangles;
ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$;
iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$.
2009 Olympic Revenge, 1
Given a scalene triangle $ABC$ with circuncenter $O$ and circumscribed circle $\Gamma$. Let $D, E ,F$ the midpoints of $BC, AC, AB$. Let $M=OE \cap AD$, $N=OF \cap AD$ and $P=CM \cap BN$. Let $X=AO \cap PE$, $Y=AP \cap OF$. Let $r$ the tangent of $\Gamma$ through $A$. Prove that $r, EF, XY$ are concurrent.