Found problems: 85335
2005 Iran MO (3rd Round), 3
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
2009 Tournament Of Towns, 5
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?
[i](5 points)[/i]
2009 May Olympiad, 2
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities.
Remember that the number $1$ is not prime.
2001 China National Olympiad, 1
Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively.
Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.
2021-IMOC, N4
There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours.
Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies
of $n$.
[i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]
2020 USMCA, 4
Suppose $n > 1$ is an odd integer satisfying $n \mid 2^{\frac{n-1}{2}} + 1$. Prove [color=red]or disprove[/color] that $n$ is prime.
[i] Note: unfortunately, the original form of this problem did not include the red text, rendering it unsolvable. We sincerely apologize for this error and are taking concrete steps to prevent similar issues from reoccurring, including computer-verifying problems where possible. All teams will receive full credit for the question.[/i]
1974 Bulgaria National Olympiad, Problem 5
Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$, $CA$, $AB$ is maximal.
[i]H. Lesov[/i]
Today's calculation of integrals, 886
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
2000 CentroAmerican, 2
Determine all positive integers $ n$ such that it is possible to tile a $ 15 \times n$ board with pieces shaped like this:
[asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]
2008 Poland - Second Round, 3
We have a positive integer $ n$ such that $ n \neq 3k$. Prove that there exists a positive integer $ m$ such that $ \forall_{k\in N \ k\geq m} \ k$ can be represented as a sum of digits of some multiplication of $ n$.
2008 AMC 12/AHSME, 9
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
1994 Portugal MO, 1
Determine the smallest natural number that has exactly $1994$ divisors.
2023 Tuymaada Olympiad, 1
Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$
1979 IMO Shortlist, 16
Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.
2009 VTRMC, Problem 4
Two circles $\alpha,\beta$ touch externally at the point $X$. Let $A,P$ be two distinct points on $\alpha$ different from $X$, and let $AX$ and $PX$ meet $\beta$ again in the points $B$ and $Q$ respectively. Prove that $AP$ is parallel to $QB$.
2010 Contests, 2
Jane has two bags $X$ and $Y$. Bag $X$ contains 4 red marbles and 5 blue marbles (and nothing else). Bag $Y$ contains 7 red marbles and 6 blue marbles (and nothing else). Jane will choose one of her bags at random (each bag being equally likely). From her chosen bag, she will then select one of the marbles at random (each marble in that bag being equally likely). What is the probability that she will select a red marble?
2007 Estonia Math Open Junior Contests, 9
In an exam with k questions, n students are taking part. A student fails the exam
if he answers correctly less than half of all questions. Call a question easy if more than half of all students answer it correctly. For which pairs (k, n) of positive integers is it possible that
(a) all students fail the exam although all questions are easy;
(b) no student fails the exam although no question is easy?
1960 Polish MO Finals, 1
Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.
1985 Miklós Schweitzer, 4
[b]4.[/b] Call a subset $S$ of the set $\{1,\dots,n\}$ [i]exceptional[/i] if any pair of distinct elements of $S$ are coprime. Consider an exceptional set with a maximal sum of elements (among all exceptional sets for a fixed $n$). Prove that if $n$ is sufficiently large, then each element of $S$ has at most two distinct prime divisors. ([b]N.17[/b])
[P. Erdos]
1993 Irish Math Olympiad, 1
Show that among any five points $ P_1,...,P_5$ with integer coordinates in the plane, there exists at least one pair $ (P_i,P_j)$, with $ i \not\equal{} j$ such that the segment $ P_i P_j$ contains a point $ Q$ with integer coordinates other than $ P_i, P_j$.
2011 ELMO Shortlist, 5
Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where
a) $f(n)=n\ln{n}$;
b) $f(n)=n$.
[i]David Yang.[/i]
2002 Junior Balkan MO, 1
The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.
1967 Miklós Schweitzer, 8
Suppose that a bounded subset $ S$ of the plane is a union of congruent, homothetic, closed triangles. Show that the boundary of $ S$ can be covered by a finite number of rectifiable arcs.
[i]L. Geher[/i]
1965 All Russian Mathematical Olympiad, 063
Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.
2021 EGMO, 2
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.
Here, $\mathbb{Q}$ denotes the set of rational numbers.