Found problems: 85335
2018 Poland - Second Round, 5
Let $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.
2009 JBMO Shortlist, 5
Let ${A, B, C}$ and ${O}$ be four points in plane, such that $\angle ABC>{{90}^{{}^\circ }}$ and ${OA=OB=OC}$.Define the point ${D\in AB}$ and the line ${l}$ such that ${D\in l, AC\perp DC}$ and ${l\perp AO}$. Line ${l}$ cuts ${AC}$at ${E}$ and circumcircle of ${ABC}$ at ${F}$. Prove that the circumcircles of triangles ${BEF}$and ${CFD}$are tangent at ${F}$.
1991 IMTS, 5
Two people, $A$ and $B$, play the following game with a deck of 32 cards. With $A$ starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. Eventually all of the cards are chosen, and the person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy?
1998 AMC 8, 5
Which of the following numbers is largest?
$ \text{(A)}\ 9.12344\qquad\text{(B)}\ 9.123\overline{4}\qquad\text{(C)}\ 9.12\overline{34}\qquad\text{(D)}\ 9.1\overline{234}\qquad\text{(E)}\ 9.\overline{1234} $
2017 EGMO, 5
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
1987 Traian Lălescu, 2.3
Let be a cube $ ABCDA'BC'D' $ such that $ AB=1, $ and let $ M,N,P,Q $ be points on the segments $ A'B',C'D',A'D', $ respectively, $ BC, $ excluding their extremities.
[b]a)[/b] If $ MN $ is perpendicular to $ PQ, $ then $ AM+A'P+CQ+CN=3. $
[b]b)[/b] If $ MN $ and $ PQ $ are concurrent, then $ AM\cdot CQ=A'P\cdot CN. $
1977 All Soviet Union Mathematical Olympiad, 248
Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$. The following condition is valid: $$(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn \,\,\,\, (*)$$ Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.
1973 IMO, 2
$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.
2018 Saudi Arabia BMO TST, 1
Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$
1958 Poland - Second Round, 2
Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?
2020 Greece JBMO TST, 3
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.
2017 Romania National Olympiad, 4
Let be a function $ f $ of class $ \mathcal{C}^1[a,b] $ whose derivative is positive. Prove that there exists a real number $ c\in (a,b) $ such that
$$ f(f(b))-f(f(a))=(f'(c))^2(b-a) . $$
2008 Romania National Olympiad, 3
Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.
2024-IMOC, N8
Find all integers $(a,b)$ satisfying: there is an integer $k>1$ such that
$$a^k+b^k-1\ |\ a^n+b^n-1$$
holds for all integer $n\geq k$ (we define that $0|0$)
2023-IMOC, N2
Find all pairs of positive integers $(a, b)$ such that $a^b+b^a=a!+b^2+ab+1$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Solve the equation $$16x^3 = (11x^2 + x -1)\sqrt{x^2 - x + 1}.$$
2003 Bosnia and Herzegovina Team Selection Test, 1
Board has written numbers: $5$, $7$ and $9$. In every step we do the following: for every pair $(a,b)$, $a>b$ numbers from the board, we also write the number $5a-4b$. Is it possible that after some iterations, $2003$ occurs at the board ?
2011 Math Prize For Girls Problems, 3
The figure below shows a triangle $ABC$ with a semicircle on each of its three sides.
[asy]
unitsize(5);
pair A = (0, 20 * 21) / 29.0;
pair B = (-20^2, 0) / 29.0;
pair C = (21^2, 0) / 29.0;
draw(A -- B -- C -- cycle);
label("$A$", A, S);
label("$B$", B, S);
label("$C$", C, S);
filldraw(arc((A + C)/2, C, A)--cycle, gray);
filldraw(arc((B + C)/2, C, A)--cycle, white);
filldraw(arc((A + B)/2, A, B)--cycle, gray);
filldraw(arc((B + C)/2, A, B)--cycle, white);
[/asy]
If $AB = 20$, $AC = 21$, and $BC = 29$, what is the area of the shaded region?
2016 Saudi Arabia Pre-TST, 1.3
A lock has $16$ keys arranged in a $4\times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too. Show that no matter what the starting positions are, it is always
possible to open this lock. (Only one key at a time can be switched.)
1978 Romania Team Selection Test, 6
[b]a)[/b] Prove that $ 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \} $
[b]b)[/b] Prove that there exist three positive rational numbers $ a,b,c $ such that the expression $ E(x,y,z):=xa+yb+zc $ vanishes for infinitely many integer triples $ (x,y,z), $ but it doesn´t get arbitrarily close to $ 0. $
2015 Math Prize for Girls Problems, 19
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
2009 Cono Sur Olympiad, 4
Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. First Andrea divides the table in $33$ zones. Each zone is formed by $3$ contiguous cells aligned vertically or horizontally, as shown in the figure.
[code]
._
|_|
|_| _ _ _
|_| |_|_|_|
[/code]
Then, Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell in such a way that the sum of the numbers in each zone is equal to $5$. Bruno wins if the sum of the numbers written in each of the $9$ columns of the table is a prime number. Otherwise, Andrea wins. Show that Bruno always has a winning strategy.
2022 Olimphíada, 4
Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and:
$$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$
Find the value of $a_{2022!}$.
1959 Czech and Slovak Olympiad III A, 2
Let $a, b, c$ be real numbers such that $a+b+c > 0$, $ab+bc+ca > 0$, $abc > 0$. Show that $a, b, c$ are all positive.
2007 Poland - Second Round, 2
We are given a cyclic quadrilateral $ABCD \quad AB\not=CD$. Quadrilaterals $AKDL$ and $CMBN$ are rhombuses with equal sides. Prove, that $KLMN$ is cyclic