Found problems: 85335
2006 Bosnia and Herzegovina Team Selection Test, 5
Triangle $ABC$ is inscribed in circle with center $O$. Let $P$ be a point on arc $AB$ which does not contain point $C$. Perpendicular from point $P$ on line $BO$ intersects side $AB$ in point $S$, and side $BC$ in $T$. Perpendicular from point $P$ on line $AO$ intersects side $AB$ in point $Q$, and side $AC$ in $R$.
(i) Prove that triangle $PQS$ is isosceles
(ii) Prove that $\frac{PQ}{QR}=\frac{ST}{PQ}$
2015 Caucasus Mathematical Olympiad, 5
Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?
2014 Estonia Team Selection Test, 4
In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.
1977 Vietnam National Olympiad, 1
Find all real $x$ such that $ \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}> \frac{x - 1}{x}$
2023 Math Prize for Girls Problems, 4
Let $\triangle A_1A_2A_3$ be an equilateral triangle with unit side length. For $k = 1$, $2$, and $3$, let $B_k$ be the point on the boundary of $\triangle A_1A_2A_3$ located $1/3$ unit away from $A_k$ in the clockwise direction and let $C_k$ be the point on the boundary of $\triangle A_1A_2A_3$ located $1/3$ unit away from $A_k$ in the counterclockwise direction. What fraction of the area of $\triangle A_1A_2A_3$ is the area of the intersection of $\triangle B_1B_2B_3$ and $\triangle C_1C_2C_3$?
2020 LMT Fall, 3
Circles $C_1,C_2,$ and $C_3$ have radii $2,3,$ and $6$ respectively. If the fourth circle $C_4$ is the sum of the areas of $C_1,C_2,$ and $C_3,$ compute the radius of $C_4.$
[i]Proposed by Alex Li[/i]
1971 IMO Longlists, 11
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.
2011 IFYM, Sozopol, 8
Let $S$ be the set of all 9-digit natural numbers, which are written only with the digits 1, 2, and 3. Find all functions $f:S\rightarrow \{1,2,3\}$ which satisfy the following conditions:
(1) $f(111111111)=1$, $f(222222222)=2$, $f(333333333)=3$, $f(122222222)=1$;
(2) If $x,y\in S$ differ in each digit position, then $f(x)\neq f(y)$.
2010 AMC 10, 13
Angelina drove at an average rate of $ 80$ kph and then stopped $ 20$ minutes for gas. After the stop, she drove at an average rate of $ 100$ kph. Altogether she drove $ 250$ km in a total trip time of $ 3$ hours including the stop. Which equation could be used to solve for the time $ t$ in hours that she drove before her stop?
$ \textbf{(A)}\ 80t\plus{}100(8/3\minus{}t)\equal{}250 \qquad
\textbf{(B)}\ 80t\equal{}250 \qquad
\textbf{(C)}\ 100t\equal{}250 \\
\textbf{(D)}\ 90t\equal{}250 \qquad
\textbf{(E)}\ 80(8/3\minus{}t)\plus{}100t\equal{}250$
1984 AMC 12/AHSME, 15
If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is
A. $18^\circ$
B. $30^\circ$
C. $36^\circ$
D. $45^\circ$
E. $60^\circ$
2016 AMC 10, 18
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$
2004 Purple Comet Problems, 3
In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$?
[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC;
draw(A--B--C--cycle);
for(int i = 1; i < 4; ++i) {
AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4);
draw(AB[i-1] -- AC[i-1]);
}
filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7));
label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]
2001 India IMO Training Camp, 1
For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.
2019 Junior Balkan MO, 4
A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.
1971 IMO Longlists, 42
Show that for nonnegative real numbers $a,b$ and integers $n\ge 2$,
\[\frac{a^n+b^n}{2}\ge\left(\frac{a+b}{2}\right)^n\]
When does equality hold?
2019 Cono Sur Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.
2014 IPhOO, 12
A circular loop with radius $r$ spins with angular frequency $\omega$ in a generated magnetic field of strength $B$. It is hooked to a resistor load $R$. How much work is done by the generator in one revolution?
[i]Problem proposed by Ahaan Rungta[/i]
1988 AMC 8, 9
An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?
[asy]
for(int a=0; a<12; ++a)
{
draw((a,0)--(a,6));
}
for(int b=0; b<7; ++b)
{
draw((0,b)--(11,b));
}
draw((0,6)--(2,6)--(1,4)--cycle,linewidth(1));
draw((3,4)--(3,6)--(5,4)--cycle,linewidth(1));
draw((0,1)--(3,2)--(6,1)--cycle,linewidth(1));
draw((7,4)--(6,6)--(9,4)--cycle,linewidth(1));
draw((8,1)--(9,3)--(10,0)--cycle,linewidth(1));[/asy]
$ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5 $
1995 Bundeswettbewerb Mathematik, 2
Let $S$ be a union of finitely many disjoint subintervals of $[0,1]$ such that no two points in $S$ have distance $1/10$. Show that the total length of the intervals comprising $S$ is at most $1/2$.
2016 ASDAN Math Tournament, 6
For what positive value $k$ does the equation $\ln x=kx^2$ have exactly one solution?
2009 Purple Comet Problems, 9
Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?
2010 HMNT, 1
$16$ progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has $2$ players remaining?
1996 Mexico National Olympiad, 2
There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?
1987 Greece Junior Math Olympiad, 1
We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance $1$ among them.
1967 German National Olympiad, 3
Prove the following theorem:
If $n > 2$ is a natural number, $a_1, ..., a_n$ are positive real numbers and becomes $\sum_{i=1}^n a_i = s$, then the following holds
$$\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}$$