Found problems: 85335
Given is an acute triangle $ABC$ with $BC < CA < AB$. Points $K$ and $L$ lie on segments $AC$ and $AB$ and satisfy $AK = AL = BC$. Perpendicular bisectors of segments $CK$ and $BL$ intersect line $BC$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at $M$. Prove that $CK + KM = BL + LM$.
Points $A,B,C$, and $M$ are given in the plane.
(a) Let $D$ be the point in the plane such that $DA\le CA$ and $DB\le CB$. Prove that there exists point $N$ satisfying $NA\le MA,NB\le MB$, and $ND\le MC$.
(b) Let $A',B',C'$ be the points in the plane such that $A'B'\le AB,A'C'\le AC,B'C'\le BC$. Does there exist a point $M'$ which satisfies the inequalities $M'A'\le MA,M'B'\le MB,M'C'\le MC$?
To whom is the discovery of the nuclear atom attributed?
$ \textbf{(A) }\text{Neils Bohr}\qquad\textbf{(B) }\text{Louis deBroglie}\qquad$
$\textbf{(C) }\text{Robert Millikan}\qquad\textbf{(D) }\text{Ernest Rutherford}\qquad $
Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$
The numbers $1, 2, 3, \ldots , 10$ are written on the blackboard. In each step, Andrew chooses two numbers $a, b$ which are written on the blackboard such that $a\geqslant 2b$, he erases them, and in their place writes the number $a-2b$.
Find all numbers $n$, such that after a sequence of steps as above, at the end only the number $n$ will remain on the blackboard.
Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear.
[i](Proposed by Tan Rui Xuen)[/i]
On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?
A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
Let $n$ be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of $n.$
(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities.
(ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$.
(iii) Give a rational number between $11/24$ and $6/13$.
(iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10.
(v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]
A natural number is a [i]factorion[/i] if it is the sum of the factorials of each of its decimal digits. For example, $145$ is a factorion because $145 = 1! + 4! + 5!$.
Find every 3-digit number which is a factorion.
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence
$a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence
$ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.
A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?
Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$ where, $s=a_1+a_2+..+a_n$
Prove that if in the tetrahedron $ ABCD $ we have $ AB = CD $, $ AC = BD $, $ AD = BC $, then all faces of the tetrahedron are acute-angled triangles.
A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?
Prove that any integer has a multiple consisting of all ten digits $\{0,1,2,3,4,5,6,7,8,9\}$.
\\
[i]Note: Any digit can be repeated any number of times[/i]
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.
[i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:
1) $ f(i,i)=0$ for all integers $0\leq i \leq n$
2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$, $j$, $k$, $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$.
Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$
Let $n$ be a positive integer. Prove that there exist an infinity of multiples of $n$ which do not contain the digit “$9$” in their decimal representation
Are there positive integers $a$, $b$, $c$ and $d$ such that:
a) $a^{2021} + b^{2023} = 11(c^{2022} + d^{2024})$;
b) $a^{2022} + b^{2022} = 11(c^{2022} + d^{2022})$?
The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.
Let $n$ be a natural number and $\alpha ,\beta ,\gamma$ be the angles of an acute triangle. Determine the least possible value of the sum:
$T=tan^n \alpha+tan^n \beta+tan^n \gamma$.