Found problems: 85335
KoMaL A Problems 2017/2018, A. 709
Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality$$\displaystyle
C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$$holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots <x_n$ of real numbers.
2016 CMIMC, 8
Consider the sequence of sets defined by $S_0=\{0,1\},S_1=\{0,1,2\}$, and for $n\ge2$, \[S_n=S_{n-1}\cup\{2^n+x\mid x\in S_{n-2}\}.\] For example, $S_2=\{0,1,2\}\cup\{2^2+0,2^2+1\}=\{0,1,2,4,5\}$. Find the $200$th smallest element of $S_{2016}$.
1970 IMO Longlists, 18
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
1998 Croatia National Olympiad, Problem 1
Which number is greater:
$$A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},$$where each of the numbers above contains $1998$ zeros?
2023 IFYM, Sozopol, 1
On the board, the numbers from $1$ to $n$ are written. Achka (A) and Bavachka (B) play the following game. First, A erases one number, then B erases two consecutive numbers, then A erases three consecutive numbers, and finally B erases four consecutive numbers. What is the smallest $n$ such that B can definitely make her moves, no matter how A plays?
2023 LMT Fall, 2
Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{7}$
Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games.
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2023 OMpD, 3
Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements.
[b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.
2019 MMATHS, 1
$S$ is a set of positive integers with the following properties:
(a) There are exactly $3$ positive integers missing from $S$.
(b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow a and b to be the same.)
Find all possibilities for the set $S$ (with proof).
2017 AIME Problems, 10
Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i, $ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
1990 Tournament Of Towns, (259) 3
A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case.
(D. Fomin, Leningrad)
2007 Estonia Math Open Junior Contests, 2
The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.
2000 Croatia National Olympiad, Problem 4
Let $S$ be the set of all squarefree numbers and $n$ be a natural number. Prove that
$$\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.$$
1994 IMO Shortlist, 3
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
2013 Moldova Team Selection Test, 1
For any positive real numbers $x,y,z$, prove that
$\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$
2017 Puerto Rico Team Selection Test, 4
Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is:
i) $3 \times 3$
ii) $3 \times 4$
2017 ASDAN Math Tournament, 6
You roll three six-sided dice. If the three dice and indistinguishable, how many combinations of numbers can result?
1990 Greece National Olympiad, 4
Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$
Estonia Open Junior - geometry, 2018.1.5
Let $M$ be the intersection of the diagonals of a cyclic quadrilateral $ABCD$. Find the length of $AD$, if it is known that $AB=2$ mm , $BC = 5$ mm, $AM = 4$ mm, and $\frac{CD}{CM}= 0.6$.
2022 Austrian MO National Competition, 6
(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$.
(b) Show that a corresponding decomposition into $30$ squares is also possible.
[i](Walther Janous)[/i]
1991 ITAMO, 1
For every triangle $ABC$ inscribed in a circle $\Gamma$ , let $A',B',C'$ be the intersections of the bisectors of the angles at $A,B,C$ with $\Gamma$ . Consider the triangle $A'B'C'$ .
(a) Do triangles $A'B'C'$ go over all possible triangles inscribed in $\Gamma$ as $\vartriangle ABC$ varies? If not, what are the constraints?
(b) Prove that the angle bisectors of $\vartriangle ABC$ are the altitudes of $\vartriangle A',B',C'$ .
2015 Puerto Rico Team Selection Test, 2
In the triangle $ABC$, let $P$, $Q$, and $R$ lie on the sides $BC$, $AC$, and $AB$ respectively, such that $AQ = AR$, $BP = BR$ and $CP = CQ$. Let $\angle PQR=75^o$ and $\angle PRQ=35^o$. Calculate the measures of the angles of the triangle $ABC$.
Oliforum Contest IV 2013, 6
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
LMT Speed Rounds, 2016.22
Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
[i]Proposed by Nathan Ramesh
2025 Serbia Team Selection Test for the IMO 2025, 2
Let $ABC$ be an acute triangle. Let $A'$ be the reflection of point $A$ over the line $BC$. Let $O$ and $H$ be the circumcenter and the orthocenter of triangle $ABC$, respectively, and let $E$ be the midpoint of segment $OH$. Let $D$ and $L$ be the points where the reflection of line $AA'$ with respect to line $OA'$ intersects the circumcircle of triangle $ABC$, where point $D$ lies on the arc $BC$ not containing $A$. If \( M \) is a point on the line \( BC \) such that \( OM \perp AD \), prove that \( \angle MAD = \angle EAL \).
[i]Proposed by Strahinja Gvozdić[/i]
2005 IberoAmerican, 3
Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.