Olympiad problems

2025 Belarusian National Olympiad, 8.3

Tags: number theory

A positive integer with three digits is written on the board. Each second the number $n$ on the board gets replaced by $n+\frac{n}{p}$, where $p$ is the largest prime divisor of $n$. Prove that either after 999 seconds or 1000 second the number on the board will be a power of two. [i]A. Voidelevich[/i]

2016 Saint Petersburg Mathematical Olympiad, 7

Tags: algebra , polynomial , Integer Polynomial , Integer

A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.

IV Soros Olympiad 1997 - 98 (Russia), 11.9

Tags: algebra , inequalities

The numbers $a$, $b$ and $c$ satisfy the conditions $$0 < a \le b \le c\,\,\,,\,\,\, a+b+ c = 7\,\,\,, \,\,\,abc = 9.$$ Within what limits can each of the numbers $a$, $b$ and $c$ vary?

2016 Harvard-MIT Mathematics Tournament, 33

Tags:

$\textbf{(Lucas Numbers)}$ The Lucas numbers are defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \ge 0$. There are $N$ integers $1 \le n \le 2016$ such that $L_n$ contains the digit $1$. Estimate $N$. An estimate of $E$ earns $\left\lfloor 20 - 2|N-E| \right\rfloor$ or $0$ points, whichever is greater.

2020 Princeton University Math Competition, A3/B5

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and radius $10$. Let sides $AB$, $BC$, $CD$, and $DA$ have midpoints $M, N, P$, and $Q$, respectively. If $MP = NQ$ and $OM + OP = 16$, then what is the area of triangle $\vartriangle OAB$?

2011 Thailand Mathematical Olympiad, 8

Tags:

Given $\Delta ABC$ and its centroid $G$, If line $AC$ is tangent to $\odot (ABG)$. Prove that, \begin{align*} AB+BC \leq 2AC \end{align*}

1999 AMC 12/AHSME, 26

Tags: geometry , perimeter , calculus , integration

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$

2019 May Olympiad, 5

Tags: combinatorics

There is a board with three rows and $2019$ columns. In the first row are written the numbers integers from $1$ to $2019$ inclusive, ordered from smallest to largest. In the second row, $Ana$ writes those same numbers but ordered at your choice. In each box in the third row write the difference between the two numbers already written in the same column (the largest minus the smallest). $Beto$ have to paint some numbers in the third row so that the sum of the numbers painted is equal to the sum of the numbers in that row that were left unpainted. Can $Ana$ complete the second row so that $Beto$ does not achieve his goal?

2019 AMC 10, 5

Tags: geometry , geometric transformation , reflection , AMC , AMC 10 , AMC 10 B , analytic geometry

Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is [u][i]not[/i][/u] always true? $(A)$ Triangle $A'B'C'$ lies in the first quadrant. $(B)$ Triangles $ABC$ and $A'B'C'$ have the same area. $(C)$ The slope of line $AA'$ is $-1$. $(D)$ The slopes of lines $AA'$ and $CC'$ are the same. $(E)$ Lines $AB$ and $A'B'$ are perpendicular to each other.

Denmark (Mohr) - geometry, 2000.2

Tags: geometry , 3D geometry , sphere , prism , hexagon

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

1996 Tournament Of Towns, (493) 6

Tags: geometry , angles

In an equilateral triangle $ABC$, let $D$ be a point on the side $AB$ such that $AD = AB /n$. Prove that the sum of $n - 1$ angles $\angle DP_lA$, $\angle DP_2A$, $...$, $\angle DP_nA$ where $P_1$, $P_2$, $...$ ,$P_{n-1}$ are the points dividing the side $BC$ into $n$ equal parts, is equal to $30$ degrees if (a) $n = 3$ (b) $n$ is an arbitrary integer, $n > 2$. (V Proizvolov)

2023/2024 Tournament of Towns, 7

Tags: combinatorics

On the table there are $2n$ coins that look the same. It is known that $n$ of them weigh 9 g. each, while the remaining $n$ weigh 10 g. each. It is required to split the coins into $n$ pairs with total weight of each pair 19 g. Prove that this can be done in less than $n$ weighings using a balance without additional weights (the balance shows which pan is heavier or that their weight is equal).

2000 South africa National Olympiad, 4

Tags: geometry , perimeter , trigonometry , function , inequalities , analytic geometry , angle bisector

$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.

2011 Morocco National Olympiad, 2

Tags: algebra unsolved , algebra

Solve in $\mathbb{R}$ the equation : $(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.

1970 Putnam, A5

Tags: Putnam , circle , ellipsoid

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$

2019 Balkan MO, 1

Tags: number theory , BMO , BMO Shortlist

Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that: $$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$ holds for all $p,q\in\mathbb{P}$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

1897 Eotvos Mathematical Competition, 3

Tags: construction , geometry

Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersections of some line $e$ with the sides $AB, CD$ and $AD, BC$, respectively (or their extensions). Given the points $M, N, P, Q$ and the length $p$ of side $AB$, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?

1961 Leningrad Math Olympiad, grade 7

Tags: algebra , geometry , combinatorics , number theory , leningrad math olympiad

[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. [b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively [b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b] [b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img] [b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

2005 AMC 10, 22

Tags: factorial

For how many positive integers $ n$ less than or equal to $ 24$ is $ n!$ evenly divisible by $ 1 \plus{} 2 \plus{} \dots \plus{} n$? $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 21$

2019 Turkey EGMO TST, 6

Tags: combinatorics , combinatorics proposed

There are $k$ piles and there are $2019$ stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are $k$ piles left and all of them contain different number of stonws. Find the maximum of $k$.

1964 All Russian Mathematical Olympiad, 045

Tags: geometry , hexagon

a) Given a convex hexagon $ABCDEF$ with all the equal angles. Prove that $$|AB|-|DE| = |EF|-|BC| = |CD|-|FA|$$ b) The opposite problem: Prove that it is possible to construct a convex hexagon with equal angles of six segments $a_1,a_2,...,a_6$, whose lengths satisfy the condition $$a_1-a_4 = a_5-a_2 = a_3-a_6$$