Found problems: 85335
2020 Costa Rica - Final Round, 4
Consider the function $ h$, defined for all positive real numbers, such that:
$$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$
for all $x > 0$. Find $h(x)$ and the value of $h(4)$.
2023 Turkey Team Selection Test, 1
Let $ABCD$ be a trapezoid with $AB \parallel CD$. A point $T$ which is inside the trapezoid satisfies $ \angle ATD = \angle CTB$. Let line $AT$ intersects circumcircle of $ACD$ at $K$ and line $BT$ intersects circumcircle of $BCD$ at $L$.($K \neq A$ , $L \neq B$) Prove that $KL \parallel AB$.
2007 Nordic, 3
The number $10^{2007}$ is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations:
1) replace a number $x$ on the blackboard with two integers $a,b>1$ such that $ab=x$.
2) strike off one or both of two equal numbers on the blackboard.
The person who cannot perform any operation loses.
Who has the winning strategy if Anne starts?
2005 Iran MO (3rd Round), 1
Find all $n,p,q\in \mathbb N$ that:\[2^n+n^2=3^p7^q\]
1996 Austrian-Polish Competition, 3
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
2020 IOM, 4
Given three positive real numbers $a,b,c$ such that following holds $a^2=b^2+bc$, $b^2=c^2+ac$
Prove that $\frac{1}{c}=\frac{1}{a}+\frac{1}{b}$.
2005 Sharygin Geometry Olympiad, 3
Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.
2009 Princeton University Math Competition, 4
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
2019 BMT Spring, Tie 4
Consider a regular triangular pyramid with base $\vartriangle ABC$ and apex $D$. If we have $AB = BC =AC = 6$ and $AD = BD = CD = 4$, calculate the surface area of the circumsphere of the pyramid.
2013 Online Math Open Problems, 6
Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$.
[i]Proposed by Michael Kural[/i]
1973 AMC 12/AHSME, 14
Each valve $ A$, $ B$, and $ C$, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $ A$ and $ C$ open it takes 1.5 hours, and with only valves $ B$ and $ C$ open it takes 2 hours. The number of hours required with only valves $ A$ and $ B$ open is
$ \textbf{(A)}\ 1.1 \qquad
\textbf{(B)}\ 1.15 \qquad
\textbf{(C)}\ 1.2 \qquad
\textbf{(D)}\ 1.25 \qquad
\textbf{(E)}\ 1.75$
2000 Chile National Olympiad, 7
Consider the following equation in $x$: $$ax (x^2 + ax + 1) = b (x^2 + b + 1).$$ It is known that $a, b$ are real such that $ab <0$ and furthermore the equation has exactly two integer roots positive. Prove that under these conditions $a^2 + b^2$ is not a prime number.
2021 AMC 12/AHSME Spring, 24
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
2017 Czech-Polish-Slovak Junior Match, 1
Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits
$a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$.
2023 Chile TST IMO, 3
Solve the system of equations in real numbers:
\[
\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x}
\]
\[
x^2 + y^2 + z^2 = 294
\]
\[
x + y + z = 0
\]
1997 IberoAmerican, 3
Let $n \geq2$ be an integer number and $D_n$ the set of all the points $(x,y)$ in the plane such that its coordinates are integer numbers with: $-n \le x \le n$ and $-n \le y \le n$.
(a) There are three possible colors in which the points of $D_n$ are painted with (each point has a unique color). Show that with
any distribution of the colors, there are always two points of $D_n$ with the same color such that the line that contains them does not go through any other point of $D_n$.
(b) Find a way to paint the points of $D_n$ with 4 colors such that if a line contains exactly two points of $D_n$, then, this points have different colors.
2007 QEDMO 5th, 1
Let $ a$, $ b$ and $ k$ be three positive integers.
We define two sequences $ \left( a_{n}\right)$ and $ \left( b_{n}\right)$ by the starting values $ a_{1}\equal{}a$ and $ b_{1}\equal{}b$ and the recurrent equations $ a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n}$ and $ b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n}$ for each positive integer $ n$.
Prove that if $ a_{1}\perp b_{1}$, $ a_{2}\perp b_{2}$ and $ a_{3}\perp b_{3}$ hold, then $ a_{n}\perp b_{n}$ holds for every positive integer $ n$.
Here, the abbreviation $ x\perp y$ stands for "the numbers $ x$ and $ y$ are coprime".
2012 Romania National Olympiad, 3
[color=darkred]Let $A,B\in\mathcal{M}_4(\mathbb{R})$ such that $AB=BA$ and $\det (A^2+AB+B^2)=0$ . Prove that:
\[\det (A+B)+3\det (A-B)=6\det (A)+6\det (B)\ .\][/color]
2000 Baltic Way, 16
Prove that for all positive real numbers $a,b,c$ we have
\[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]
2013 Vietnam Team Selection Test, 3
Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:
i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$;
ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$.
Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$.
a) Proof that $|F|=\infty$.
b) Proof that for each $f\in F$ then $|v(f)|<\infty$.
c) Find the maximum value of $|v(f)|$ for $f\in F$.
2012 239 Open Mathematical Olympiad, 4
For some positive numbers $a$, $b$, $c$ and $d$, we know that
$$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$
Prove that
$$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$
1954 AMC 12/AHSME, 1
The square of $ 5\minus{}\sqrt{y^2\minus{}25}$ is:
$ \textbf{(A)}\ y^2\minus{}5\sqrt{y^2\minus{}25} \qquad
\textbf{(B)}\ \minus{}y^2 \qquad
\textbf{(C)}\ y^2 \\
\textbf{(D)}\ (5\minus{}y)^2 \qquad
\textbf{(E)}\ y^2\minus{}10\sqrt{y^2\minus{}25}$
1963 Czech and Slovak Olympiad III A, 3
A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.
2000 AMC 8, 15
Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?
[asy]
pair A,B,C,D,EE,F,G;
A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3));
EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2);
draw(A--B--C--cycle);
draw(D--EE--A);
draw(EE--F--G);
label("$A$",A,S);
label("$B$",B,SW);
label("$C$",C,N);
label("$D$",D,NE);
label("$E$",EE,NE);
label("$F$",F,SE);
label("$G$",G,SE);
[/asy]
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$
2016 Iran Team Selection Test, 2
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.