Found problems: 85335
2012 Putnam, 5
Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]
2022 Romania EGMO TST, P1
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]
2008 National Chemistry Olympiad, 1
Which element is a liquid at $25^\circ\text{C}$ and $1.0 \text{ atm}$?
$\textbf{(A)}\hspace{.05in}\text{bromine} \qquad\textbf{(B)}\hspace{.05in}\text{krypton} \qquad\textbf{(C)}\hspace{.05in}\text{phosphorus} \qquad\textbf{(D)}\hspace{.05in}\text{xenon} \qquad$
1993 Turkey MO (2nd round), 6
$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$.
Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)
2016 Moldova Team Selection Test, 9
Let $\alpha \in \left( 0, \dfrac{\pi}{2}\right)$.Find the minimum value of the expression
$$ P = (1+\cos\alpha)\left(1+\frac{1}{\sin \alpha} \right)+(1+\sin \alpha)\left(1+\frac{1}{\cos \alpha} \right) .$$
2008 Purple Comet Problems, 6
Three friends who live in the same apartment building leave the building at the same time to go rock climbing on a cliff just outside of town. Susan walks to the cliff and climbs to the top of the cliff. Fred runs to the cliff twice as fast as Susan walks and climbs the top of the cliff at a rate that is only two-thirds as fast as Susan climbs. Ralph bikes to the cliff at a speed twice as fast as Fred runs and takes two hours longer to climb to the top of the cliff than Susan does. If all three friends reach the top of the cliff at the same time, how many minutes after they left home is that?
2018 Purple Comet Problems, 22
Positive integers $a$ and $b$ satisfy $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$. Find $a^2 + b^2 + c^2$.
2015 Tuymaada Olympiad, 5
There is some natural number $n>1$ on the board. Operation is adding to number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by $3^{2000}$
[i]A. Golovanov[/i]
2013 Sharygin Geometry Olympiad, 6
Dear Mathlinkers,
1. A, B the end points of an arch circle
2. (O) a circle tangent to AB intersecting the arch in question
3. T the point of contact of (O) and AB
4. C, D the points of intersection of (O) with the arch in the order A, D, C, B
5. E, F the points of intersection of AC and DT, BD and CT.
Prove : EF is parallel to AB.
Sincerely
Jean-Louis
1985 Miklós Schweitzer, 11
Let $\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let
$$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$
be the tangent bundle of $E$, where $V\xi=\mathrm{Ker}\, d\pi$ is the vertical subbundle of $\tau_E$. Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$. Construct a linear connection $\nabla$ on $V\xi$ such that
$$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$
($X$ and $Y$ are vector fields on $E$, $[.,\, .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$. [J. Szilasi]
2017 China Team Selection Test, 2
Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.
1992 Turkey Team Selection Test, 1
The feet of perpendiculars from the intersection point of the diagonals of cyclic quadrilateral $ABCD$ to the sides $AB,BC,CD,DA$ are $P,Q,R,S$, respectively. Prove $PQ+RS=QR+SP$.
Durer Math Competition CD Finals - geometry, 2015.D1
From all three vertices of triangle $ABC$, we set perpendiculars to the exterior and interior of the other vertices angle bisectors. Prove that the sum of the squares of the segments thus obtained is exactly $2 (a^2 + b^2 + c^2)$, where $a, b$, and $c$ denote the lengths of the sides of the triangle.
2004 Putnam, B4
Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.
2015 IMO Shortlist, C5
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:
(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.
Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$.
[i]Proposed by Ivan Guo and Ross Atkins, Australia[/i]
2013 International Zhautykov Olympiad, 3
Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that
\[\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.\]
[i]Proposed by Orif Ibrogimov, Uzbekistan.[/i]
2014 Sharygin Geometry Olympiad, 2
A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.
(A. Zertsalov, D. Skrobot)
2022 Thailand TSTST, 2
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2018 Singapore Senior Math Olympiad, 2
In a convex quadrilateral $ABCD, \angle A < 90^o, \angle B < 90^o$ and $AB > CD$. Points $P$ and $Q$ are on the segments $BC$ and $AD$ respectively. Suppose the triangles $APD$ and $BQC$ are similar. Prove that $AB$ is parallel to $CD$.
2022 AIME Problems, 2
Find the three-digit positive integer $\underline{a} \ \underline{b} \ \underline{c}$ whose representation in base nine is $\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}$, where $a$, $b$, and $c$ are (not necessarily distinct) digits.
1991 AMC 12/AHSME, 26
An $n$-digit positive integer is [i]cute[/i] if its $n$ digits are an arrangement of the set $\{1,2,\ldots,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,\ldots,n$. For example 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4 $
2010 Today's Calculation Of Integral, 663
Given are the curve $y=x^2+x-2$ and a curve which is obtained by tranfering the curve symmetric with respect to the point $(p,\ 2p)$. Let $p$ change in such a way that these two curves intersects, find the maximum area of the part bounded by these curves.
[i]1978 Nagasaki University entrance exam/Economics[/i]
2023-24 IOQM India, 22
In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective opposite vertex by a line segment. If $N$ denotes the number of ways we can choose the pegs such that the drawn line segments divide the interior of the triangle into exactly nine regions, find the sum of the squares of the digits of $N$.
PEN S Problems, 27
Which integers have the following property? If the final digit is deleted, the integer is divisible by the new number.
2016 Czech-Polish-Slovak Junior Match, 6
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia