Found problems: 85335
Geometry Mathley 2011-12, 15.4
Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$.
Kostas Vittas
2020 Junior Balkаn MO, 1
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2005 AMC 12/AHSME, 10
A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2010 Argentina Team Selection Test, 6
Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$.
Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$.
Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).
2013 Bulgaria National Olympiad, 4
Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$.
Prove that:
\[\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0\]
[i]Proposed by Nikolay Nikolov[/i]
1982 IMO Shortlist, 4
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
2001 China Team Selection Test, 3
Given sets $A = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$, $B = \{2, 3, 8, 10, 12, 13, 14, 15, 18\}$, if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to $A$ when divided by 19, then that positive integer is called an $\alpha$ number. If a positive integer leaves a remainder that belongs to $B$ when divided by 19, then that positive integer is called a $\beta$ number.
(1) For what positive integer $n$, among all its positive divisors, are the numbers of $\alpha$ divisors and $\beta$ divisors equal?
(2) For which positive integers $k$, are the numbers of $\alpha$ divisors less than the numbers of $\beta$ divisors? For which positive integers $l$, are the numbers of $\alpha$ divisors greater than the numbers of $\beta$ divisors?
2013 Gheorghe Vranceanu, 2
Given two natural numbers $ n\ge 2,a, $ prove that there exists another natural number $ v\ge 2 $ such that:
$$ \frac{v+\sqrt{v^2-4}}{2} =\left( \frac{n+\sqrt{n^2-4}}{2} \right)^a $$
2007 AMC 12/AHSME, 5
Last year Mr. John Q. Public received an inheritance. He paid $ 20\%$ in federal taxes on the inheritance, and paid $ 10\%$ of what he had left in state taxes. He paid a total of $ \$10,500$ for both taxes. How many dollars was the inheritance?
$ \textbf{(A)}\ 30,000 \qquad \textbf{(B)}\ 32,500 \qquad \textbf{(C)}\ 35,000 \qquad \textbf{(D)}\ 37,500 \qquad \textbf{(E)}\ 40,000$
1985 ITAMO, 15
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
2020 Tournament Of Towns, 3
Let $ABCD$ be a rhombus, let $APQC$ be a parallelogram such that the point $B$ lies inside it and the side $AP$ is equal to the side of the rhombus. Prove that $B$ is the orthocenter of the triangle $DPQ$.
Egor Bakaev
2017 ASDAN Math Tournament, 2
Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$?
1984 Brazil National Olympiad, 4
$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?
1953 Putnam, B2
Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$
1990 China Team Selection Test, 2
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$
2013 Purple Comet Problems, 5
A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border is a rectangle twice as long as it is wide. How many inches long is that rectangle?
2011 Today's Calculation Of Integral, 741
Evaluate
\[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]
2004 IMO Shortlist, 8
For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$.
[i]Proposed by Marcin Kuczma, Poland [/i]
2017 Iranian Geometry Olympiad, 4
$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Can the number of [b]clockwise[/b] triangles be exactly $2017$?
[i]Proposed by Morteza Saghafian[/i]
1984 Iran MO (2nd round), 6
Let $D$ and $D'$ be two lines with the equations
\[\frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \quad \text{and} \quad \frac{x+1}{2} = \frac{y+2}{4} = \frac{z-1}{3}.\]
Find the length of their common perpendicular.
2014 Dutch Mathematical Olympiad, 2 juniors
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, G, H;
A = (0,0);
B = (2,0);
D = (0.5,1.5);
C = B + D - A;
G = reflect(A,B)*(C) + C - B;
H = reflect(B,C)*(H) + A - B;
draw(H--A--D--C--G);
draw(interp(A,G,-0.1)--interp(A,G,1.1));
draw(interp(C,H,-0.1)--interp(C,H,1.1));
draw(D--G--H--cycle, dashed);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, E);
dot("$D$", D, NW);
dot("$G$", G, NE);
dot("$H$", H, SE);
[/asy]
2019 Durer Math Competition Finals, 15
$ABC$ is an isosceles triangle such that $AB = AC$ and $\angle BAC = 96^o$. $D$ is the point for which $\angle ACD = 48^o$, $AD = BC$ and triangle $DAC$ is obtuse-angled. Find $\angle DAC$.
2019 SAFEST Olympiad, 5
There are $25$ IMO participants attending a party. Every two of them speak to each other in some language, and they use only one language even if they both know some other language as well. Among every three participants there is a person who uses the same language to speak to the other two (in that group of three). Prove that there is an IMO participant who speaks the same language to at least $10$ other participants
2006 China Northern MO, 4
Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.
2021 Novosibirsk Oral Olympiad in Geometry, 2
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.