Found problems: 85335
2001 Saint Petersburg Mathematical Olympiad, 10.1
Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq
\sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$
[I]Proposed by A. Khrabrov[/i]
2014 Contests, 1
Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.)
[i]Robert Simson et al.[/i]
2023 Romania EGMO TST, P4
Let $n\geqslant 3$ be an integer and $a_1,\ldots,a_n$ be nonzero real numbers, with sum $S{}$. Prove that \[\sum_{i=1}^n\left|\frac{S-a_i}{a_i}\right|\geqslant\frac{n-1}{n-2}.\]
2004 AMC 12/AHSME, 2
In the expression $ c\cdot a^b\minus{}d$, the values of $ a$, $ b$, $ c$, and $ d$ are $ 0$, $ 1$, $ 2$, and $ 3$, although not necessarily in that order. What is the maximum possible value of the result?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 6\qquad
\textbf{(C)}\ 8\qquad
\textbf{(D)}\ 9\qquad
\textbf{(E)}\ 10$
VI Soros Olympiad 1999 - 2000 (Russia), 11.2
Find the sum of all possible products of natural numbers of the form $k_1k_2...k_{999}$, where in each product $ k_1 < k_2 < ... < k_{999} <1999$, and there are no $k_i$ and $k_j$ such that $k_i + k_j =1999$.
2011 District Olympiad, 2
[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle.
[b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication:
$$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$
2020 Miklós Schweitzer, 4
Consider horizontal and vertical segments in the plane that may intersect each other. Let $n$ denote their total number. Suppose that we have $m$ curves starting from the origin that are pairwise disjoint except for their endpoints. Assume that each curve intersects exactly two of the segments, a different pair for each curve. Prove that $m=O(n)$.
2022 Bulgarian Autumn Math Competition, Problem 8.1
Solve the equation:
\[4x^2+|9-6x|=|10x-15|+6(2x+1)\]
2020 South Africa National Olympiad, 4
A positive integer $k$ is said to be [i]visionary[/i] if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?
1975 USAMO, 3
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.
2020 AMC 12/AHSME, 15
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$
2024 Dutch IMO TST, 2
Let $ABC$ be a triangle. A point $P$ lies on the segment $BC$ such that the circle with diameter $BP$ passes through the incenter of $ABC$. Show that $\frac{BP}{PC}=\frac{c}{s-c}$ where $c$ is the length of segment $AB$ and $2s$ is the perimeter of $ABC$.
2001 USA Team Selection Test, 4
There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.
2013 AMC 10, 9
Three positive integers are each greater than $1$, have a product of $27000$, and are pairwise relatively prime. What is their sum?
$\textbf{(A) }100\qquad \textbf{(B) } 137\qquad\textbf{(C) } 156\qquad\textbf{(D) }160\qquad\textbf{(E) }165$
Kyiv City MO Seniors 2003+ geometry, 2016.10.4
On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line.
(Igor Nagel)
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 8
Let $ f(x) \equal{} x \minus{} \frac {1}{x}.$ How many different solutions are there to the equation $ f(f(f(x))) \equal{} 1$?
A. 1
B. 2
C. 3
D. 6
E. 8
2023 Oral Moscow Geometry Olympiad, 3
In an acute triangle $ABC$ the line $OI$ is parallel to side $BC$. Prove that the center of the nine-point circle of triangle $ABC$ lies on the line $MI$, where $M$ is the midpoint of $BC$.
2002 Iran Team Selection Test, 12
We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic.
[i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.
2007 Estonia Math Open Senior Contests, 2
Three circles with centres A, B, C touch each other pairwise externally, and touch circle c from inside. Prove that if the centre of c coincideswith the orthocentre of triangle ABC, then ABC is equilateral.
1957 Moscow Mathematical Olympiad, 366
Solve the system: $$\begin{cases} \dfrac{2x_1^2}{1+x_1^2}=x_2 \\ \\ \dfrac{2x_2^2}{1+x_2^2}=x_3\\ \\ \dfrac{2x_3^2}{1+x_3^2}=x_1\end{cases}$$
2005 Iran MO (3rd Round), 3
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
MathLinks Contest 3rd, 2
Let n be a positive integer and let $a_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n}$ be $4n - 1$ positive real numbers such that $c^2_{i+j} \ge a_ib_j $, for all $1 \le i, j \le n$. Also let $m = \max_{2 \le i\le 2n} c_i$.
Prove that $$\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{
b_1 + b_2 + ...+ b_n}{n}\right)$$
2021 IMO Shortlist, G4
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
2005 Today's Calculation Of Integral, 1
Calculate the following indefinite integral.
[1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$
[2] $\int \sin x\cos 3x dx$
[3] $\int \sin 2x\sin 3x dx$
[4] $\int \frac{dx}{4x^2-12x+9}$
[5] $\int \cos ^4 x dx$
2016 Bulgaria JBMO TST, 2
The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.