Found problems: 85335
2016 Costa Rica - Final Round, F1
Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.
2011 District Olympiad, 3
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function with the property that $ (f\circ f) (x) =[x], $ for any real number $ x. $ Show that there exist two distinct real numbers $ a,b $ so that $ |f(a)-f(b)|\ge |a-b|. $
$ [] $ denotes the integer part.
2023 Moldova EGMO TST, 11
Find all three digit positive integers that have distinct digits and after their greatest digit is switched to $1$ become multiples of $30$.
Sri Lankan Mathematics Challenge Competition 2022, P4
[b]Problem 4[/b] : A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent line to both circles touches the circle with $AC$ as diameter at $P \neq C$ and the circle with $CB$ as diameter at $Q \neq C.$ Prove that lines $AP, BQ$ and the common tangent line to both circles at $C$ all meet at a single point which lies on the circle with $AB$ as diameter.
2003 ITAMO, 6
Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$. The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all $n$ for which he can select the guests in such an order to serve all the guests.
2012 Tournament of Towns, 6
A bank has one million clients, one of whom is Inspector Gadget. Each client has a unique PIN number consisting of six digits. Dr. Claw has a list of all the clients. He is able to break into the account of any client, choose any $n$ digits of the PIN number and copy them. The n digits he copies from different clients need not be in the same $n$ positions. He can break into the account of each client, but only once. What is the smallest value of $n$ which allows Dr.Claw to determine the complete PIN number of Inspector Gadget?
PEN S Problems, 2
It is given that $2^{333}$ is a $101$-digit number whose first digit is $1$. How many of the numbers $2^k$, $1 \le k \le 332$, have first digit $4$?
1997 Tournament Of Towns, (561) 2
Which of the following statements are true?
(a) If a polygon can be divided into two congruent polygons by a broken line segment, it can be divided into two congruent polygons by a straight line segment.
(b) If a convex polygon can be divided into two congruent polygons by a broken line segment, it can be so divided by a straight line segment.
(c) If a convex polygon can be divided into two polygons by a broken line segment, one of which can be mapped onto the other by a combination of rotations and translations, it can be so divided by a straight line segment.
(S Markelov,)
2014 European Mathematical Cup, 3
Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram
[i]Proposed by Steve Dinh[/i]
1974 Miklós Schweitzer, 10
Let $ \mu$ and $ \nu$ be two probability measures on the Borel sets of the plane. Prove that there are random variables $ \xi_1, \xi_2, \eta_1, \eta_2$ such that
(a) the distribution of $ (\xi_1, \xi_2)$ is $ \mu$ and the distribution of $ (\eta_1, \eta_2)$ is $ \nu$,
(b) $ \xi_1 \leq \eta_1, \xi_2 \leq \eta_2$ almost everywhere, if an only if $ \mu(G) \geq \nu(G)$ for all sets of the form $ G\equal{}\cup_{i\equal{}1}^k (\minus{}\infty, x_i) \times (\minus{}\infty, y_i).$
[i]P. Major[/i]
1986 IMO Longlists, 7
Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$
[i]Simplified version.[/i]
Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$
2017 VTRMC, 1
Determine the number of real solutions to the equation $\sqrt{2 -x^2} = \sqrt[3]{3 -x^3}.$
2013 Stanford Mathematics Tournament, 7
Find all real values of $u$ such that the curves $y=x^2+u$ and $y=\sqrt{x-u}$ intersect in exactly one point.
2019 AMC 12/AHSME, 20
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$?
$\textbf{(A)} \frac{1}{3} \qquad \textbf{(B)} \frac{7}{16} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{9}{16} \qquad \textbf{(E)} \frac{2}{3}$
2023 Iberoamerican, 2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that:
$$2023f(f(x))+2022x^2=2022f(x)+2023[f(x)]^2+1$$
for each integer $x$.
1986 IMO Longlists, 22
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$
2020 Brazil Cono Sur TST, 4
A flea is, initially, in the point, which the coordinate is $1$, in the real line. At each second, from the coordinate $a$, the flea can jump to the coordinate point $a+2$ or to the coordinate point $\frac{a}{2}$. Determine the quantity of distinct positions(including the initial position) which the flea can be in until $n$ seconds.
For instance, if $n=1$, the flea can be in the coordinate points $1,3$ or $\frac{1}{2}$.
1991 Arnold's Trivium, 63
Is there a solution of the Cauchy problem $y\partial u/\partial x+\sin x\partial u/\partial y=y$, $u|_{x=0}=y^4$ on the whole $(x,y)$ plane? Is it unique?
1976 Miklós Schweitzer, 7
Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent.
[i]L. Lempert[/i]
2020/2021 Tournament of Towns, P2
There were ten points $X_1, \ldots , X_{10}$ on a line in this particular order. Pete constructed an isosceles triangle on each segment $X_1X_2, X_2X_3,\ldots, X_9X_{10}$ as a base with the angle $\alpha{}$ at its apex. It so happened that all the apexes of those triangles lie on a common semicircle with diameter $X_1X_{10}$. Find $\alpha{}$.
[i]Egor Bakaev[/i]
2013 IMC, 2
Let $\displaystyle{p,q}$ be relatively prime positive integers. Prove that
\[\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}\]
[i]Proposed by Alexander Bolbot, State University, Novosibirsk.[/i]
1994 AMC 12/AHSME, 26
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw(unitsquare);
path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;
draw(p);
draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);
draw(shift((0,-2-sqrt(2)))*p);
draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 $
2010 Belarus Team Selection Test, 3.2
Prove that there exists a positive integer $n$ such that $n^6 + 31n^4 - 900\vdots 2009 \cdot 2010 \cdot 2011$.
(I. Losev, I. Voronovich)
2014 China Team Selection Test, 1
Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$).
Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.
2007 Moldova Team Selection Test, 2
If $I$ is the incenter of a triangle $ABC$ and $R$ is the radius of its circumcircle then \[AI+BI+CI\leq 3R\]