Found problems: 85335
Kvant 2021, M2665
The polynomials $f(x)$ and $g(x)$ are given. The points $A_1(f(1),g(1)),\ldots,A_n(f(n),g(n))$ are marked on the coordinate plane. It turns out that $A_1\ldots A_n$ is a regular $n{}$-gon. Prove that the degree of at least one of $f{}$ and $g{}$ is at least $n-1$.
[i]Proposed by V. Bragin[/i]
2019 Simon Marais Mathematical Competition, B2
For each odd prime number $p$, prove that the integer
$$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$
(Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)
2023 BMT, 9
Let triangle $\vartriangle ABC$ be acute, and let point $M$ be the midpoint of $\overline{BC}$. Let $E$ be on line segment $\overline{AB}$ such that $\overline{AE} \perp \overline{EC}$. Then, suppose $T$ is a point on the other side of $\overleftrightarrow{BC}$ as $A$ is such that $\angle BTM = \angle ABC$ and $\angle TCA = \angle BMT$. If $AT = 14$, $AM = 9,$ and $\frac{AE}{AC} =\frac27$ , compute $BC$.
2018 Taiwan TST Round 2, 1
Let $A,B,C$ be the midpoints of the three sides $B'C', C'A', A'B'$ of the triangle $A'B'C'$ respectively. Let $P$ be a point inside $\Delta ABC$, and $AP,BP,CP$ intersect with $BC, CA, AB$ at $P_a,P_b,P_c$, respectively. Lines $P_aP_b, P_aP_c$ intersect with $B'C'$ at $R_b, R_c$ respectively, lines $P_bP_c, P_bP_a$ intersect with $C'A'$ at $S_c, S_a$ respectively. and lines $P_cP_a, P_cP_b$ intersect with $A'B'$ at $T_a, T_b$, respectively. Given that $S_c,S_a, T_a, T_b$ are all on a circle centered at $O$.
Show that $OR_b=OR_c$.
2010 IMAC Arhimede, 5
Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$
2005 Abels Math Contest (Norwegian MO), 2a
In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.
2006 Croatia Team Selection Test, 3
Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.
2014 South East Mathematical Olympiad, 1
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$
2011 Morocco National Olympiad, 2
Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system :
$\left\{\begin{matrix}
x+y+z+t=4\\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt}
\end{matrix}\right.$
2007 Turkey MO (2nd round), 3
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that
$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
2010 Contests, 2
Show that
\[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \]
for all positive real numbers $a, \: b, \: c.$
2016 Hong Kong TST, 5
Let $ABCD$ be inscribed in a circle with center $O$. Let $E$ be the intersection of $AC$ and $BD$. $M$ and $N$ are the midpoints of the arcs $AB$ and $CD$ respectively (the arcs not containing any other vertices). Let $P$ be the intersection point of $EO$ and $MN$. Suppose $BC=5$, $AC=11$, $BD=12$, and $AD=10$. Find $\frac{MN}{NP}$
1997 Romania National Olympiad, 1
function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon:
1- $f(0,x)=x+1$
2- $f(x+1,0)=f(x,1)$
3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997)
find $f(3,1997)$
2014 Purple Comet Problems, 11
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.
2021 Romania Team Selection Test, 3
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$
2023 Belarusian National Olympiad, 10.2
A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one.
Find the maximum possible number of distinct prime divisors of $n$
2007 Sharygin Geometry Olympiad, 5
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to:
a) five?
b) four?
2003 Purple Comet Problems, 10
How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol?
2000 India Regional Mathematical Olympiad, 3
Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]
2005 Today's Calculation Of Integral, 6
Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$
MBMT Team Rounds, 2020.30
Let the number of ways for a rook to return to its original square on a $4\times 4$ chessboard in 8 moves if it starts on a corner be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path.
[i]Proposed by Bradley Guo[/i]
2012 Federal Competition For Advanced Students, Part 1, 4
Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$.
Show that $\gamma$ is the second-largest angle in the triangle $ABC$.
2021 LMT Spring, A8
Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$.
[i]Proposed by Powell Zhang[/i]
2023 Stars of Mathematics, 2
Let $a{}$ and $b{}$ be positive integers, whose difference is a prime number. Prove that $(a^n+a+1)(b^n+b+1)$ is not a perfect square for infinitely many positive integers $n{}$.
[i]Proposed by Vlad Matei[/i]
2010 AMC 10, 20
Two circles lie outside regular hexagon $ ABCDEF$. The first is tangent to $ \overline{AB}$, and the second is tangent to $ \overline{DE}$. Both are tangent to lines $ BC$ and $ FA$. What is the ratio of the area of the second circle to that of the first circle?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 81\qquad\textbf{(E)}\ 108$