Found problems: 167
Kvant 2019, M2586
A polygon is given in which any two adjacent sides are perpendicular. We call its two vertices non-friendly if the bisectors of the polygon emerging from these vertices are perpendicular. Prove that for any vertex the number of vertices that are not friends with it is even.
Kvant 2021, M2680
Let $n>1$ be a natural number and $A_0A_1\ldots A_{2^n-2}$ be a regular polygon. Prove that \[\frac{1}{A_0A_1}=\frac{1}{A_0A_2}+\frac{1}{A_0A_4}+\frac{1}{A_0A_8}+\cdots+\frac{1}{A_0A_{2^{n-1}}}.\][i]Proposed by Le Hoang and Ngoc Thai (Vietnam)[/i]
2021/2022 Tournament of Towns, P6
There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob?
[i]Alexandr Gribalko[/i]
Kvant 2022, M2690
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.
[i]Proposed by A. Antropov[/i]
2019 Tournament Of Towns, 5
In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?
Kvant 2023, M2732
Let $O{}$ be the circumcenter of the triangle $ABC$. On the rays $AC$ and $BC$ consider the points $C_a$ and $C_b$ respectively, such that $AC_a$ and $BC_b$ are equal in length to $AB$. Let $O_c{}$ be the circumcenter of the triangle $CC_aC_b$. Define the points $O_a{}$ and $O_b{}$ similarly. Prove that $O{}$ is the orthocenter of the triangle $O_aO_bO_c$.
[i]Proposed by A. Zaslavsky[/i]
Kvant 2020, M2621
Consider a triangle $ABC$ in which $AB<BC<CA$. The excircles touch the sides $BC, CA,$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. A circle is drawn through the points $A_1, B_1$ and $C_1$ which intersects the sides $BC, CA$ and $AB$ for the second time at the points $A_2, B_2$ and $C_2$ respectively. On which side of the triangle can lie the largest of the segments $A_1A_2, B_1B_2$ and $C_1C_2$?
[i]Proposed by I. Weinstein[/i]
Kvant 2022, M2720
Let $\Omega$ be the circumcircle of the triangle $ABC$. The points $M_a,M_b$ and $M_c$ are the midpoints of the sides $BC, CA$ and $AB{}$, respectively. Let $A_l, B_l$ and $C_l$ be the intersection points of $\Omega$ with the rays $M_cM_b, M_aM_c$ and $M_bM_a$ respectively. Similarly, let $A_r, B_r$ and $C_r$ be the intersection points of $\Omega$ with the rays $M_bM_c, M_cM_a$ and $M_aM_b$ respectively. Prove that the mean of the areas of the triangles $A_lB_lC_l$ and $A_rB_rC_r$ is not less than the area of the triangle $ABC$.
[i]Proposed by L. Shatunov and T. Kazantseva[/i]
Kvant 2021, M2681
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$.
[i]Proposed by I. Dorofeev[/i]
Kvant 2022, M2729
Determine all positive integers $n{}$ and $m{}$ such that $m^n=n^{3m}$.
[i]Proposed by I. Dorofeev[/i]
Kvant 2020, M2629
The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that
[list=a]
[*]all colored quadrilaterals are trapezoids;
[*]the areas of all polygons of the same color are equal;
[*]the ratios of the bases of one-color trapezoids are equal;
[*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied.
[/list]
[i]Proposed by F. Nilov[/i]
[center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]
Kvant 2022, M2687
We have a regular $n{}$-gon, with $n\geqslant 4$. We consider the arrangements of $n{}$ numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement $K{}$, we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by $\alpha(K)$.
[list=a]
[*]Find the largest possible value of $\alpha(K)$.
[*]Find the number of arrangements for which $\alpha(K)$ takes this largest possible value.
[/list]
[i]Proposed by P. Kozhevnikov[/i]
Kvant 2020, M233
Two digits one are written at the ends of a segment. In the middle, their sum is written, the number 2. Then, in the middle between each two neighboring numbers written, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written?
[i]Proposed by G. Halperin[/i]
Kvant 2020, M2626
An infinite number of participants gathered for the Olympiad, who were registered under the numbers $1, 2,\ldots$. It turns out that for every $n = 1, 2,\ldots$ a participant with number $n{}$ has at least $n{}$ friends among the remaining participants (note: friendship is mutual). There is a hotel with an infinite number of double rooms. Prove that the participants can be accommodated in double rooms so that there is a couple of friends in each room.
[i]Proposed by V. Bragin, P. Kozhevnikov[/i]
Kvant 2021, M2644
Petya and Vasya are playing on an $100\times 100$ board. Initially, all the cells of the board are white. With each of his moves, Petya paints one or more white cells standing on the same diagonal in black. With each of his moves, Vasya paints one or more white cells standing on the same column in black. Petya makes the first move. The one who can't make a move loses. Who has a winning strategy?
[i]Proposed by M. Didin[/i]
Kvant 2019, M2578
Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers
\[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \]
are integers. Prove that $p=q=r $.
[i]Nazar Agakhanov[/i]
Kvant 2020, M2614
In an $n\times n$ table, it is allowed to rearrange rows, as well as rearrange columns. Asterisks are placed in some $k{}$ cells of the table. What maximum $k{}$ for which it is always possible to ensure that all the asterisks are on the same side of the main diagonal (and that there are no asterisks on the main diagonal itself)?
[i]Proposed by P. Kozhevnikov[/i]
Kvant 2019, M2566
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
Kvant 2020, M2620
A satellite is considered accessible from the point $A{}$ of the planet's surface if it is located relative to the tangent plane drawn at point $A{}$, strictly on the other side than the planet. What is the smallest number of satellites that need to be launched over a spherical planet so that at some point the signals of at least two satellites are available from each point on the planet's surface?
[i]Proposed by S. Volchenkov[/i]
Kvant 2021, M2636
We call a natural number $p{}$ [i]simple[/i] if for any natural number $k{}$ such that $2\leqslant k\leqslant \sqrt{p}$ the inequality $\{p/k\}\geqslant 0,01$ holds. Is the set of simple prime numbers finite?
[i]Proposed by M. Didin[/i]
Kvant 2020, M1
In a country, the time for presidential elections has approached. There are exactly 20 million voters in the country, of which only one percent supports the current president, Miraflores. Naturally, he wants to be elected again, but on the other hand, he wants the elections to seem democratic.
Miraflores established the following voting process: all the voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, and so on. In the smallest groups, a representative is chosen. Then, the chosen electors choose representatives in the second-smallest groups, to vote in an even larger group, and so on. Finally, the representatives of the largest groups choose the president.
Miraflores divides voters into groups as he wants and instructs his supporters how to vote. Will he be able to organize the elections in such a way that he will be elected president? (If the votes are equal, the opposition wins.)
[i]From the 32nd Moscow Mathematical Olympiad[/i]
Kvant 2020, M600
Two cyclists ride on two intersecting circles. Each of them rides on his own circle at a constant speed. Having left at the same time from one of the points of intersection of the circles and having made one lap each, the cyclists meet again at this point. Prove that there exists a fixed point in the plane, the distances from which to cyclists are the same all the time, regardless of the directions they travel in.
[i]Proposed by N. Vasiliev and I. Sharygin[/i]
Kvant 2020, M2590
In an acute triangle $ABC$ the point $O{}$ is the circumcenter, $H_1$ is the foot of the perpendicular from $A{}$ onto $BC$, and $M_H$ and $N_H$ are the projections of $H_1$ on $AC$ and $AB{}$, respectively. Prove that the polyline $M_HON_H$ divides the triangle $ABC$ in two figures of equal area.
[i]Proposed by I. A. Kushner[/i]
Kvant 2019, M2555
In each cell of a $2019\times 2019$ board is written the number $1$ or the number $-1$. Prove that for some positive integer $k$ it is possible to select $k$ rows and $k$ columns so that the absolute value of the sum of the $k^2$ numbers in the cells at the intersection of the selected rows and columns is more than $1000$.
[i]Folklore[/i]
Kvant 2019, M2571
Let $ABCD$ be a trapezoid with $AD \parallel BC$, $AD < BC$. Let $E$ be a point on the side $AB$ and $F$ be point on the side $CD$. The circle $(AEF)$ intersects the segment $AD$ again at $A_1$ and the circle $(CEF)$ intersects these segment $BC$ again at $C_1$. Prove that the lines $A_1 C_1$, $BD$ and $EF$ are concurrent.
[i]Proposed by A. Kuznetsov[/i]