Found problems: 85335
There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that:
[list=a]
[*]one of its angles is not greater than $60^\circ{}$;
[*]one of its angles is at least $120^\circ$.
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[i]Maxim Didin[/i]
All empty white triangles in figure are to be filled with integers such that for each gray triangle the three numbers in the white neighboring triangles sum to a multiple of $5$. The lower left and the lower right white triangle are already filled with the numbers $12$ and $3$, respectively.
Find all integers that can occur in the uppermost white triangle.
(G. Woeginger, Eindhoven, The Netherlands)
[img]https://cdn.artofproblemsolving.com/attachments/8/a/764732f5debbd58a147e9067e83ba8d31f7ee9.png[/img]
If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]
Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.
It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?
Prove for all real numbers $x, y$, $$(x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \geq 0.$$ Determine when equality holds.
Prove that if we erase $n-3$ diagonals of a regular $n$-gon, then we may still choose $n-3$ of the remaining diagonals such that they don't intersect inside the $n$-gon; but it is possible to erase $n-2$ diagonals such that this statement doesn't hold.
We are given coins of $1,2,5,10,20,50$ lipas and of $1$ kuna (Croatian currency: $1$ kuna = $100$ lipas). Prove that if a bill of $M$ lipas can be paid by $N$ coins, then a bill of $N$ kunas can be paid by M coins.
\[\lim_{n\to\infty} n^2 \int_0^1 x^n e^{-x}\log(x)\,\mathrm dx\]
[i]Proposed by Connor Gordon and Vlad Oleksenko[/i]
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.
[i]Tiger Zhang[/i]
Find all natural numbers $ n\ge 3 $ and real numbers $ a $ which have the property that the polynomial $ X^n-aX-1 $ admits a monic quadratic integer polynomial.
[i]Mihai Bălună[/i]
A class consisting of $24$ students has participated in the first round of the Georg Mohr Contest, where one could obtain between $0$ and $20$ points. Three of the students obtained exactly the class’s average. If each of the students that scored below the average had scored $4$ points more, the average would have been $3$ points higher. How many students scored above the class’s average?
The number of real roots of the equation $\sin x=\lg x$ is
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}$more than $3$
Let $M$ be a set of real $n \times n$ matrices such that
i) $I_{n} \in M$, where $I_n$ is the identity matrix.
ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both
iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$.
iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$.
Prove that $M$ contains at most $n^2 $ matrices.
Find all positive integers $n$ for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that
$$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$
[i]Proposed by Ali Partofard[/i]
There are some markets in a city. Some of them are joined by one-way streets, such that for any market there are exactly two streets to leave it. Prove that the city may be partitioned into $1014$ districts such that streets join only markets from different districts, and by the same one-way for any two districts (either only from first to second, or vice-versa).
Given a polygon $ABCDEFGHIJ$.
How many diagonals does the polygon have?
Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that
\[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$.
[i]Victor Wang.[/i]
Find all integer coefficient polynomials $Q$ such that
[list]
[*] $Q(n)\ge 1$ $\forall n\in \mathbb{Z}_+$.
[*] $Q(mn)$ and $Q(m)Q(n)$ have the same number of prime divisors $\forall m,n\in\mathbb{Z}_+$.
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Alan is bored one day and decides to write down all the divisors of $1260^2$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to erase all his work. Every second, he picks a random divisor which is still on the wall and instantly erases it and every number that divides it. What is the expected time it takes for Alan to erase everything on the wall?
Points $E$ and $F$ lying on sides $BC$ and $AD$ respectively of a parallelogram $ABCD$ are such that $EF=ED=DC$. Let $M$ be the midpoint of $BE$ and $MD$ meet $EF$ at $G$. Prove that $\angle EAC=\angle GBD$.
Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.
Given is an acute triangle $ABC$. Point $P$ lies inside this triangle and lies on the bisector of angle $\angle BAC$. Suppose that the point of intersection of the altitudes $H$ of triangle $ABP$ lies inside triangle $ABC$. Let $Q$ be the intersection of the line $AP$ and the line perpendicular to $AC$ passing through $H$. Prove that $Q$ is the point symmetrical to $P$ wrt the line $BH$.