Olympiad problems

Kvant 2021, M2635

In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$. [i]Porposed by O. Titov[/i]

Kvant 2019, M2569

Tags: combinatorics , Kvant

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

2019 Tournament Of Towns, 4

Tags: algebra , number theory , Kvant

Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.

Kvant 2022, M2695

Tags: geometry , Kvant

Let the circle $\Omega$ and the line $\ell$ intersect at two different points $A{}$ and $B{}$. For different and non-points. Let $X$ and $T$ be points on $\ell$ and $Y$ and $Z$ be points on $\Omega$, all of them different from $A{}$ and $B{}$. Prove the following statements: [list=a] [*]The points $X,Y$ and $Z$ lie on the same line if and only if \[\frac{\overline{AX}}{\overline{BX}}=\pm\frac{AY}{BY}\cdot\frac{AZ}{BZ}.\] [*]The points $X,Y,Z$ and $T$ lie on the same circle if and only if \[\frac{\overline{AX}}{\overline{BX}}\cdot\frac{\overline{AT}}{\overline{BT}}=\pm\frac{AY}{BY}\cdot\frac{AZ}{BZ}.\] [/list] Note: In both points, the sign $+$ is selected in the right parts of the equalities if the points $Y{}$ and $Z{}$ lie on the same arc $AB$ of the circle $\Omega$, and the sign $-$ if $Y{}$ and $Z{}$ lie on different arcs $AB$. By $\overline{AX}/\overline{BX}$, we indicate the ratio of the lengths of $AX$ and $BX$, taken with the sign $+$ or $-$ depending on whether the $AX$ and $BX$ vectors are co-directed or oppositely directed. [i]Proposed by M. Skopenkov[/i]

Kvant 2020, M2622

Tags: geometry , rhombus , Kvant

The points $E, F, G$ and $H{}$ are located on the sides $DA, AB, BC$ and $CD$ of the rhombus $ABCD$ respectively, so that the segments $EF$ and $GH$ touch the circle inscribed in the rhombus. Prove that $FG\parallel HE$. [i]Proposed by V. Eisenstadt[/i]

2019 Caucasus Mathematical Olympiad, 4

Tags: combinatorics , Kvant

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

Kvant 2021, M2646

Tags: combinatorics , number theory , Kvant

Koshchey opened an account at the bank. Initially, it had 0 rubles. On the first day, Koshchey puts $k>0$ rubles in, and every next day adds one ruble more there than the day before. Each time after Koshchey deposits money into the account, the total amount in the account is divided by two by the bank. Find all such $k{}$ for which the amount on the account will always be an integer number of rubles. [i]Proposed by S. Berlov[/i]

Kvant 2022, M2714

Tags: number theory , polynomial , Kvant , Divisibility

Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$. [i]From the folklore[/i]

Kvant 2020, M2607

Tags: number theory , Kvant

Let $n$ be a natural number. The set $A{}$ of natural numbers has the following property: for any natural number $m\leqslant n$ in the set $A{}$ there is a number divisible by $m{}$. What is the smallest value that the sum of all the elements of the set $A{}$ can take? [i]Proposed by A. Kuznetsov[/i]

Kvant 2020, M2597

Tags: number theory , Kvant , prime numbers

Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$. [i]Proposed by M. Antipov[/i]

Kvant 2022, M2712

Tags: Kvant , geometry , trigonometry , inequalities

Let $ABC$ be a triangle, with $\angle A=\alpha,\angle B=\beta$ and $\angle C=\gamma$. Prove that \[\sum_{\text{cyc}}\tan \frac{\alpha}{2}\tan\frac{\beta}{2}\cot\frac{\gamma}{2}\geqslant\sqrt{3}.\][i]Proposed by R. Regimov (Azerbaijan)[/i]

Kvant 2019, M2567

Tags: geometry , Kvant

On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.

Kvant 2021, M2672

Tags: geometry , Kvant

Let the inscribed circle $\omega$ of the triangle $ABC$ have a center $I{}$ and touch the sides $BC, CA$ and $AB$ at points $D, E$ and $F{}$ respectively. Let $M{}$ and $N{}$ be points on the straight line $EF$ such that $BM \parallel AC$ and $CN \parallel AB$. Let $P{}$ and $Q{}$ be points on the segments $DM{}$ and $DN{}$, respectively, such that $BP \parallel CQ$. Prove that the intersection point of the lines $PF$ and $QE$ lies on $\omega$. [i]Proposed by Don Luu (Vietnam)[/i]

2019 Tournament Of Towns, 5

Tags: tangential , cyclic quadrilateral , Cyclic , geometry , area , Kvant

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

Kvant 2019, M2546

Tags: algebra , Kvant

Let $a,b,c$ be real numbers $a + b +c = 0$. Show that [list=a] [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^3 + b^3 + c^3}{3} = \frac{a^5 + b^5 + c^5}{5}$. [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^5 + b^5 + c^5}{5} = \frac{a^7 + b^7 + c^7}{7}$. [/list] [I]Folklore[/I]

Kvant 2019, M2542

Tags: combinatorics , Kvant

A grasshopper is in the left above corner of a $10\times 10$ square. At each step he can jump a square below or a square to the right. Also, he can also fly from a cell of the bottom row to a cell of the above row, and from a cell of the rightmost column to a cell of the leftmost column. Prove that the grasshopper has to do at leat $9$ flies in order to visit each cell of the square at least once. [I]Proposed by N. Vlasova[/I]

Kvant 2019, M2559

Tags: geometry , fixed , angle bisector , Fixed point , circles , Coloring , Kvant

Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane. (Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version [url=https://artofproblemsolving.com/community/c6h2094701p15140671]here[/url]) [hide=note]There was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.[/hide]