Found problems: 85335
2024 Junior Balkan Team Selection Tests - Moldova, 3
Let $M$ be a set of 999 points in the plane with the property: For any 3 distinct points in $M$ we can choose two of them, such that the distance between them is less than $1$.
a)Prove that there exists a disc of radius not greater than 1 that covers at least 500 points in $M$.
b)Is it true that there always exists a disc of radius not greater than 1 that covers at least 501 points in $M$?
2008 South East Mathematical Olympiad, 4
Let $m, n$ be positive integers $(m, n>=2)$. Given an $n$-element set $A$ of integers $(A=\{a_1,a_2,\cdots ,a_n\})$, for each pair of elements $a_i, a_j(j>i)$, we make a difference by $a_j-a_i$. All these $C^2_n$ differences form an ascending sequence called “derived sequence” of set $A$. Let $\bar{A}$ denote the derived sequence of set $A$. Let $\bar{A}(m)$ denote the number of terms divisible by $m$ in $\bar{A}$ . Prove that $\bar{A}(m)\ge \bar{B}(m)$ where $A=\{a_1,a_2,\cdots ,a_n\}$ and $B=\{1,2,\cdots ,n\}$.
2005 AMC 12/AHSME, 11
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
$ \textbf{(A)}\ 41\qquad
\textbf{(B)}\ 42\qquad
\textbf{(C)}\ 43\qquad
\textbf{(D)}\ 44\qquad
\textbf{(E)}\ 45$
2022 Federal Competition For Advanced Students, P2, 3
Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:
The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation.
Lisa repeats this until she gets a number for the first time was on the board.
(a) Show that the sequence of moves always ends.
(b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board?
Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$.
Since Lisa gets $21$ for the second time, the turn order ends.
[i](Stephan Pfannerer)[/i]
Today's calculation of integrals, 855
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
2011 F = Ma, 12
You are given a large collection of identical heavy balls and lightweight rods. When two balls are placed at the ends of one rod and interact through their mutual gravitational attraction (as is shown on the left), the compressive force in the rod is $F$. Next, three balls and three rods are placed at the vertexes and edges of an equilateral triangle (as is shown on the right). What is the compressive force in each rod in the latter case?
[asy]
size(300);
real x=-25;
draw((x,-8)--(x,8),linewidth(6));
filldraw(Circle((x,8),2.5),grey);
filldraw(Circle((x,-8),2.5),grey);
draw((0,-8)--(0,8)--(8*sqrt(3),0)--cycle,linewidth(6));
filldraw(Circle((0,8),2.5),grey);
filldraw(Circle((0,-8),2.5),grey);
filldraw(Circle((8*sqrt(3),0),2.5),grey);
[/asy]
(A) $\frac{1}{\sqrt{3}}F$
(B) $\frac{\sqrt{3}}{2}F$
(C) $F$
(D) $\sqrt{3}F$
(E) $2F$
II Soros Olympiad 1995 - 96 (Russia), 9.9
Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.
2023 Thailand TSTST, 4
Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.
2009 All-Russian Olympiad Regional Round, 10.3
Kostya had two sets of $17$ coins: in one set all the coins were real, and in the other set there were exactly $5$ fakes (all the coins look the same; all real coins weigh the same, all fake coins also weigh the same, but it is unknown lighter or heavier than real ones). Kostya gave away one of the sets friend, and subsequently forgot which of the two sets had stayed. With the help of two weighings, can Kostya on a cup scale without weights, find out which of the two
did he give away the sets?
1987 Yugoslav Team Selection Test, Problem 3
Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)
2014-2015 SDML (High School), 14
Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?
$\text{(A) }136\qquad\text{(B) }294\qquad\text{(C) }508\qquad\text{(D) }692\qquad\text{(E) }803$
2007 Purple Comet Problems, 6
The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum.
MBMT Guts Rounds, 2015.24
In cyclic quadrilateral $ABCD$, $\angle DBC = 90^\circ$ and $\angle CAB = 30^\circ$. The diagonals of $ABCD$ meet at $E$. If $\frac{BE}{ED} = 2$ and $CD = 60$, compute $AD$. (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)
2006 France Team Selection Test, 1
Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$.
Prove that $(PS)$ and $(QR)$ are perpendicular.
2021 Romanian Master of Mathematics, 1
Let $T_1, T_2, T_3, T_4$ be pairwise distinct collinear points such that $T_2$ lies between $T_1$ and $T_3$, and $T_3$ lies between $T_2$ and $T_4$. Let $\omega_1$ be a circle through $T_1$ and $T_4$; let $\omega_2$ be the circle through $T_2$ and internally tangent to $\omega_1$ at $T_1$; let $\omega_3$ be the circle through $T_3$ and externally tangent to $\omega_2$ at $T_2$; and let $\omega_4$ be the circle through $T_4$ and externally tangent to $\omega_3$ at $T_3$. A line crosses $\omega_1$ at $P$ and $W$, $\omega_2$ at $Q$ and $R$, $\omega_3$ at $S$ and $T$, and $\omega_4$ at $U$ and $V$, the order of these points along the line being $P,Q,R,S,T,U,V,W$. Prove that $PQ + TU = RS + VW$
[i]Geza Kos, Hungary[/i]
1987 Bulgaria National Olympiad, Problem 5
Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.
2015 India IMO Training Camp, 3
There are $n\ge 2$ lamps, each with two states: $\textbf{on}$ or $\textbf{off}$. For each non-empty subset $A$ of the set of these lamps, there is a $\textit{soft-button}$ which operates on the lamps in $A$; that is, upon $\textit{operating}$ this button each of the lamps in $A$ changes its state(on to off and off to on). The buttons are identical and it is not known which button corresponds to which subset of lamps. Suppose all the lamps are off initially. Show that one can always switch all the lamps on by performing at most $2^{n-1}+1$ operations.
2010 Stanford Mathematics Tournament, 16
A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$. The wheel is perfectly circular and has radius $5$. After the three laps, how many revolutions around its axis has the wheel been turned through?
2005 National Olympiad First Round, 18
How many integers $0\leq x < 121$ are there such that $x^5+5x^2 + x + 1 \equiv 0 \pmod{121}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
1996 Estonia Team Selection Test, 2
Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.
2020 Serbian Mathematical Olympiad, Problem 1
Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.
2004 Harvard-MIT Mathematics Tournament, 5
A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of $\tfrac{2}{3}$ of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?
2021 Saudi Arabia IMO TST, 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
1995 Singapore Team Selection Test, 1
Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function.
Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$.
(i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$.
(ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.
2021 MMATHS, 5
Suppose that $a_1 = 1$, and that for all $n \ge 2$, $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$. If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$, find $k$.
[i]Proposed by Andrew Wu[/i]