Found problems: 85335
1986 National High School Mathematics League, 6
Area of $\triangle ABC$ is $\frac{1}{4}$, circumradius of $\triangle ABC$ is $1$.
Let $s=\sqrt{a}+\sqrt{b}+\sqrt{c},t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, then
$\text{(A)}s>t\qquad\text{(B)}s=t\qquad\text{(C)}s<t\qquad\text{(D)}s>t$
2021 USEMO, 6
A bagel is a loop of $2a+2b+4$ unit squares which can be obtained by cutting a concentric $a\times b$ hole out of an $(a +2)\times (b+2)$ rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length $a+2$ of the rectangle.)
Consider an infinite grid of unit square cells. For each even integer $n \ge 8$, a bakery of order $n$ is a finite set of cells $ S$ such that, for every $n$-cell bagel $B$ in the grid, there exists a congruent copy of $B$ all of whose cells are in $S$. (The copy can be translated and rotated.) We denote by $f(n)$ the smallest possible number of cells in a bakery of order $ n$.
Find a real number $\alpha$ such that, for all sufficiently large even integers $n \ge 8$, we have $$\frac{1}{100}<\frac{f (n)}{n^ {\alpha}}<100$$
[i]Proposed by Nikolai Beluhov[/i]
2019 Brazil Team Selection Test, 6
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
1957 AMC 12/AHSME, 40
If the parabola $ y \equal{} \minus{}x^2 \plus{} bx \minus{} 8$ has its vertex on the $ x$-axis, then $ b$ must be:
$ \textbf{(A)}\ \text{a positive integer}\qquad \\
\textbf{(B)}\ \text{a positive or a negative rational number}\qquad \\
\textbf{(C)}\ \text{a positive rational number}\qquad \\
\textbf{(D)}\ \text{a positive or a negative irrational number}\qquad \\
\textbf{(E)}\ \text{a negative irrational number}$
2016 JBMO Shortlist, 5
Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\]
[i]Proposed by Azerbaijan[/i]
[hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]
1983 Vietnam National Olympiad, 2
$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$
$(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$.
.
2006 AIME Problems, 8
Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}$. Let $K$ be the area of rhombus $\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$.
[asy]
size(150);defaultpen(linewidth(0.7)+fontsize(10));
draw(rotate(45)*polygon(4));
pair F=(1+sqrt(2))*dir(180), C=(1+sqrt(2))*dir(0), A=F+sqrt(2)*dir(45), E=F+sqrt(2)*dir(-45), B=C+sqrt(2)*dir(180-45), D=C+sqrt(2)*dir(45-180);
draw(F--(-1,0)^^C--(1,0)^^A--B--C--D--E--F--cycle);
pair point=origin;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$\mathcal{P}$", intersectionpoint( A--(-1,0), F--(0,1) ));
label("$\mathcal{S}$", intersectionpoint( E--(-1,0), F--(0,-1) ));
label("$\mathcal{R}$", intersectionpoint( D--(1,0), C--(0,-1) ));
label("$\mathcal{Q}$", intersectionpoint( B--(1,0), C--(0,1) ));
label("$\mathcal{T}$", point);
dot(A^^B^^C^^D^^E^^F);[/asy]
2002 Iran MO (2nd round), 3
In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$
2005 Kazakhstan National Olympiad, 1
Solve equation
\[2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|\]
2024 Belarusian National Olympiad, 8.5
Polina wrote on the first page of her notebook $n$ different positive integers. On the second page she wrote all pairwise sums of the numbers from the first page, and on the third - absolute values of pairwise differences of number from the second page. After that she kept doing same operations, i.e. on the page $2k$ she wrote all pairwise sums of numbers from page $2k-1$, and on the page $2k+1$ absolute values of differences of numbers from page $2k$. At some moment Polina noticed that there exists a number $M$ such that, no matter how long she does her operations, on every page there are always at most $M$ distinct numbers.
What is the biggest $n$ for which it is possible?
[i]M. Karpuk[/i]
2005 Indonesia Juniors, day 2
p1. Among the numbers $\frac15$ and $\frac14$ there are infinitely many fractional numbers. Find $999$ decimal numbers between $\frac15$ and $\frac14$ so that the difference between the next fractional number with the previous fraction constant.
(i.e. If $x_1, x_2, x_3, x_4,..., x_{999}$ is a fraction that meant, then $x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}$)
p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity.
[img]https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png[/img]
It is known that the area of the triangle in Figure $ 1$ is $ 1$ unit area. Find the total area of the area formed by the black triangles in figure $5$. Also find the total area of the area formed by the black triangles in the $20$th figure.
p3. For each pair of natural numbers $a$ and $b$, we define $a*b = ab + a - b$. The natural number $x$ is said to be the [i]constituent [/i] of the natural number $n$ if there is a natural number $y$ that satisfies $x*y = n$. For example, $2$ is a constituent of $6$ because there is a natural number 4 so that $2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6$. Find all constituent of $2005$.
p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing:
a. First.
b. Second.
c. Third.
d. Tenth.
p5. Given the equation $x^2 + 3y^2 = n$, where $x$ and $y$ are integers. If $n < 20$ what number is $n$, and which is the respective pair $(x,y)$ ? Show that it is impossible to solve $x^2 + 3y^2 = 8$ in integers.
2017 Bosnia And Herzegovina - Regional Olympiad, 4
It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$
2006 All-Russian Olympiad Regional Round, 11.7
Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.
1953 AMC 12/AHSME, 19
In the expression $ xy^2$, the values of $ x$ and $ y$ are each decreased $ 25\%$; the value of the expression is:
$ \textbf{(A)}\ \text{decreased } 50\% \qquad\textbf{(B)}\ \text{decreased }75\%\\
\textbf{(C)}\ \text{decreased }\frac{37}{64}\text{ of its value} \qquad\textbf{(D)}\ \text{decreased }\frac{27}{64}\text{ of its value}\\
\textbf{(E)}\ \text{none of these}$
2005 Austrian-Polish Competition, 1
For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a [i]diagonal-quadrangle[/i] of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.
1949 Putnam, B4
Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion
$$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$
are positive integers.
2016 Grand Duchy of Lithuania, 3
Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $BF = BE$ and such that $ED$ is the angle bisector of $\angle BEC$. Prove that $BD = EF$ if and only if $AF = EC$.
1955 Poland - Second Round, 4
Inside the triangle $ ABC $ a point $ P $ is given; find a point $ Q $ on the perimeter of this triangle such that the broken line $ APQ $ divides the triangle into two parts with equal areas.
2018 Iran Team Selection Test, 2
Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$,
$$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$
[i]Proposed by Morteza Saghafian[/i]
2015 Turkey Team Selection Test, 1
Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.
1947 Putnam, B3
Let $x,y$ be cartesian coordinates in the plane. $I$ denotes the line segment $1\leq x\leq 3 , y=1.$ For every point $P$ on $I$, let $P'$ denote the point that lies on the segment joining the origin to $P$ and such that the distance $P P'$ is equal to $1 \slash 100.$ As $P$ describes $I$, the point $P'$ describes a curve $C$. Which of $I$ and $C$ has greater length?
Gheorghe Țițeica 2025, P3
Let $(a_n)_{n\geq 0}$ be a sequence defined by $a_0\geq 0$ and the recurrence relation $$a_{n+1}=\frac{a_n^2-1}{n+1},$$ for all $n\geq 0$. Prove that here exists a real number $a> 0$ such that:
[list]
[*] if $a_0\geq a,$ $\lim_{n\rightarrow\infty}a_n = \infty$;
[*] if $a_0\in [0,a),$ $\lim_{n\rightarrow\infty}a_n = 0$.
2022 JBMO Shortlist, A2
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by [i]Petar Filipovski, Macedonia[/i]
2017 Serbia JBMO TST, 2
Let $x,y,z$ be positive real numbers.Prove that
$(xy^2+yz^2+zx^2)(x^2y+y^2z+z^2x)(xy+yz+zx)\geq 3(x+y+z)^2(xyz)^2.$
2009 Hanoi Open Mathematics Competitions, 11
Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements.
Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.