Found problems: 85335
1998 National Olympiad First Round, 28
How many distinct real roots does the equation $ \sqrt{x\plus{}4\sqrt{x\minus{}4} } \minus{}\sqrt{x\plus{}2\sqrt{x\minus{}1} } \equal{}1$ have?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
2007 China Team Selection Test, 1
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
1962 Putnam, A6
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $ab$ and $a+b$, and having the property that for every rational number $r$ exactly one of the following three statements is true:
$$r\in S,\;\; -r\in S,\;\;r =0.$$
Prove that $S$ is the set of all positive rational numbers.
2000 Singapore Team Selection Test, 2
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square
2015 Princeton University Math Competition, A7
Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?
1993 All-Russian Olympiad, 4
Prove that any two rectangular prisms with equal volumes can be placed in a space such that any horizontal plain that intersects one of the prisms will intersect the other forming a polygon with the same area.
2022 IFYM, Sozopol, 8
Determine the number of ordered quadruples of integers $(a,b,c,d)$ for which
$0\leq a,b,c,d\leq 36$ and $37|a^2+b^2-c^3-d^3$.
2022 AMC 8 -, 23
A $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\triangle$s in a line.
[asy]
//diagram by kante314
size(3.3cm);
defaultpen(linewidth(1));
real r = 0.37;
path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle;
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle);
draw((0,1)--(3,1)--(3,2)--(0,2)--cycle);
draw((1,0)--(1,3)--(2,3)--(2,0)--cycle);
draw(circle((3/2,5/2),1/3));
draw(circle((5/2,1/2),1/3));
draw(circle((3/2,3/2),1/3));
draw(shift(0.5,0.38) * equi);
draw(shift(1.5,0.38) * equi);
draw(shift(0.5,1.38) * equi);
draw(shift(2.5,1.38) * equi);
draw(shift(0.5,2.38) * equi);
draw(shift(2.5,2.38) * equi);
[/asy]
How many configurations will have three $\triangle$s in a line and three $\bigcirc$s in a line?
$\textbf{(A) } 39 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 78 \qquad \textbf{(D) } 84 \qquad \textbf{(E) } 96$
2012 239 Open Mathematical Olympiad, 5
On the hypotenuse $AB$ of the right-angled triangle $ABC$, a point $K$ is chosen such that $BK = BC$. Let $P$ be a point on the perpendicular line from point $K$ to the line $CK$, equidistant from the points $K$ and $B$. Also let $L$ denote the midpoint of the segment $CK$. Prove that line $AP$ is tangent to the circumcircle of the triangle $BLP$.
2025 All-Russian Olympiad, 9.5
Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis. \\
2019 CCA Math Bonanza, I7
How many permutations $\pi$ of $\left\{1,2,\ldots,7\right\}$ are there such that $\pi\left(k\right)\leq2k$ for $k=1,\ldots,7$? A permutation $\pi$ of a set $S$ is a function from $S$ to itself such that if $a\neq b$, then $\pi\left(a\right)\neq\pi\left(b\right)$. For example, $\pi\left(x\right)=x$ and $\pi\left(x\right)=8-x$ are permutations of $\left\{1,2,\ldots,7\right\}$ but $\pi\left(x\right)=1$ is not.
[i]2019 CCA Math Bonanza Individual Round #7[/i]
1958 November Putnam, A6
Let $a(x)$ and $b(x)$ be continuous functions on $[0,1]$ and let $0 \leq a(x) \leq a <1$ on that range. Under what other conditions (if any) is the solution of the equation for $u,$
$$ u= \max_{0 \leq x \leq 1} b(x) +a(x)u$$
given by
$$u = \max_{0 \leq x \leq 1} \frac{b(x)}{1-a(x)}.$$
2018 MIG, 5
Some of the values produced by two functions, $f(x)$ and $g(x)$, are shown below. Find $f(g(3))$
\begin{tabular}{c||c|c|c|c|c}
$x$ & 1 & 3 & 5 & 7 & 9 \\ \hline\hline
$f(x)$ & 3 & 7 & 9 & 13 & 17 \\ \hline
$g(x)$ & 54 & 9 & 25 & 19 & 44
\end{tabular}
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }13\qquad\textbf{(E) }17$
1993 APMO, 2
Find the total number of different integer values the function \[ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] \] takes for real numbers $x$ with $0 \leq x \leq 100$.
2004 CentroAmerican, 3
With pearls of different colours form necklaces, it is said that a necklace is [i]prime[/i] if it cannot be decomposed into strings of pearls of the same length, and equal to each other.
Let $n$ and $q$ be positive integers. Prove that the number of prime necklaces with $n$ beads, each of which has one of the $q^n$ possible colours, is equal to $n$ times the number of prime necklaces with $n^2$ pearls, each of which has one of $q$ possible colours.
Note: two necklaces are considered equal if they have the same number of pearls and you can get the same colour on both necklaces, rotating one of them to match it to the other.
Indonesia Regional MO OSP SMA - geometry, 2004.2
Triangle $ABC$ is given. The points $D, E$, and $F$ are located on the sides $BC, CA$ and $AB$ respectively so that the lines $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{ CF}=2$
2007 China Team Selection Test, 3
Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$
1994 All-Russian Olympiad Regional Round, 11.6
Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.
2012 Tuymaada Olympiad, 4
$25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants to give to each of the $24$ remaining donkeys a pot of one of six colours of the rainbow (except red), so that at least one pot of each colour is given to some donkey (but successive donkeys can receive pots of the same colour). Which of the two friends has more ways to get his plan implemented, and how many times more?
[i]Eeyore is a character in the Winnie-the-Pooh books by A. A. Milne. He is generally depicted as a pessimistic, gloomy, depressed, old grey stuffed donkey, who is a friend of the title character, Winnie-the-Pooh. His name is an onomatopoeic representation of the braying sound made by a normal donkey. Of course, Winnie-the-Pooh is a fictional anthropomorphic bear.[/i]
[i]Proposed by F. Petrov[/i]
2023 Oral Moscow Geometry Olympiad, 2
There is a square sheet of paper. How to get a rectangular sheet of paper with an aspect ratio equal to $\sqrt2$? (There are no tools, the sheet can only be bent.)
2016 VJIMC, 4
Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying
$$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$
for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e.
$$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$
2010 Contests, 3
Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.
2012 Romania National Olympiad, 2
[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that:
\[3\le |z-a|+|z-b|+|z-c|\le 4,\]
for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]
2025 Kyiv City MO Round 1, Problem 3
The diameter \( AD \) of the circumcircle of triangle \( ABC \) intersects line \( BC \) at point \( K \). Point \( D \) is reflected symmetrically with respect to point \( K \), resulting in point \( L \). On line \( AB \), a point \( F \) is chosen such that \( FL \perp AC \). Prove that \( FK \perp AD \).
[i]Proposed by Matthew Kurskyi[/i]
2009 Harvard-MIT Mathematics Tournament, 4
Let $P$ be a fourth degree polynomial, with derivative $P^\prime$, such that $P(1)=P(3)=P(5)=P^\prime (7)=0$. Find the real number $x\neq 1,3,5$ such that $P(x)=0$.