Found problems: 85335
Find the largest natural number $n$ such that any set of $n$ tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a $20 \times 20$ table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will.
[img]https://cdn.artofproblemsolving.com/attachments/b/9/0dddb25c2aa07536b711ded8363679e47972d6.png[/img]
On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$?
[i]Proposed by S. Berlov[/i]
The line $\ell$ is perpendicular to the side $AC$ of the acute triangle $ABC$ and intersects this side at point $K$, and the circumcribed circle $\vartriangle ABC$ at points $P$ and $T$ (point P on the other side of line $AC$, as the vertex $B$). Denote by $P_1$ and $T_1$ - the projections of the points $P$ and $T$ on line $AB$, with the vertices $A, B$ belong to the segment $P_1T_1$. Prove that the center of the circumscribed circle of the $\vartriangle P_1KT_1$ lies on a line containing the midline $\vartriangle ABC$, which is parallel to the side $AC$.
(Anton Trygub)
Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$
Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.
A gang of robbers took away a bag of coins from a merchant . Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies) . Prove that after one coin is removed, the number of the remaining coins is divisible by the number of robbers.
(Folklore, modified by A Shapovalov)
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.
We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin?
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”
The best parts of grandma’s $30$ cm $ \times 30$ cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
Let $ABC$ be a triangle and $P$ be any point on the side $BC$. Let $I_1$,$I_2$ be the incenters of triangles $ABP$ and $ACP$, respectively. If $D$ is the point of tangency of the incircle of $ABC$ with the side $BC$, prove that $\angle I_1DI_2 = 90^o$.
Suppose $p > 3$ is a prime number and
\[S = \sum_{2 \le i < j < k \le p-1} ijk\]
Prove that $S+1$ is divisible by $p$.
The equality $ \frac{1}{x\minus{}1}\equal{}\frac{2}{x\minus{}2}$ is satisfied by:
$ \textbf{(A)}\ \text{no real values of }x \qquad
\textbf{(B)}\ \text{either }x\equal{}1 \text{ or }x\equal{}2 \qquad
\textbf{(C)}\ \text{only }x\equal{}1 \\
\textbf{(D)}\ \text{only }x\equal{}2 \qquad
\textbf{(E)}\ \text{only }x\equal{}0$
Find all prime numbers $p$, $q$ and $r$ such that the fourth power of any of them minus one is divisible by the product of the other two. (Author: V. Senderov)
It is known that triangle $ABC$ is not isosceles with altitudes of $AA_1, BB_1$, and $CC_1$. Suppose $B_A$ and $C_A$ respectively points on $BB_1$ and $CC_1$ so that $A_1B_A$ is perpendicular on $BB_1$ and $A_1C_A$ is perpendicular on $CC_1$. Lines $B_AC_A$ and $BC$ intersect at the point $T_A$ . Define in the same way the points $T_B$ and $T_C$ . Prove that points $T_A, T_B$, and $T_C$ are collinear.
Toner Drum and Celery Hilton are both running for president. A total of $129$ million people cast their vote in a random order, with exactly $63$ million and $66$ million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page?
[i]2017 CCA Math Bonanza Individual Round #13[/i]
Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P, Q, R, S$ lie in the interiors of segments $AB, BC, CD, DA,$ respectively, such that $$\angle PDA = \angle PCB, \text{ } \angle QAB = \angle QDC, \text{ } \angle RBC = \angle RAD, \text{ and } \angle SCD = \angle SBA.$$ Let $AQ$ intersect $BS$ at $X$, and $DQ$ intersect $CS$ at $Y$. Prove that lines $PR$ and $XY$ are either parallel or coincide.
[i]Tilek Askerbekov[/i]
Let $n$ be a positive integer. There are $n$ lamps, each with a switch that changes the lamp from on to off, or from off to on, each time it is pressed. The lamps are initially all off.
You are going to press the switches in a series of rounds. In the first round, you are going to press exactly $1$ switch; in the second round, you are going to press exactly $2$ switches; and so on, so that in the $k$th round you are going to press exactly $k$ switches. In each round you will press each switch at most once. Your goal is to finish a round with all of the lamps switched on.
Determine for which $n$ you can achieve this goal.
The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them.
For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums.
(E. Barabanov)
A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]