Olympiad problems

2018 CCA Math Bonanza, I5

Tags:

Determine all positive numbers $x$ such that \[\frac{16}{x+2}+\frac{4}{x+0}+\frac{9}{x+1}+\frac{100}{x+8}=19.\] [i]2018 CCA Math Bonanza Individual Round #5[/i]

2024 Junior Balkan Team Selection Tests - Moldova, 2

Tags: number theory

Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.

A $39$-tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies \[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\] The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers $a$, $b$ (these maxima aren't necessarily achieved using the same tuple of real numbers). Find $a + b$. [i]Proposed by Evan Chang[/i]

2020 Kosovo National Mathematical Olympiad, 2

Tags: Kosovo , national olympiad , Olympiad , number theory , divisible

Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.

1986 Federal Competition For Advanced Students, P2, 6

Tags: function , algebra unsolved , algebra

Given a positive integer $ n$, find all functions $ F: \mathbb{N} \rightarrow \mathbb{R}$ such that $ F(x\plus{}y)\equal{}F(xy\minus{}n)$ whenever $ x,y \in \mathbb{N}$ satisfy $ xy>n$.

1999 Yugoslav Team Selection Test, Problem 4

Tags: number theory , arithmetic sequence

For a natural number $d$, $M_d$ denotes the set of natural numbers which are not representable as the sum of at least two consecutive terms of an arithmetic progression with the common difference d whose terms are integers. Prove that each $c\in M_3$ can be written in the form $c=ab$, where $a\in M_1$ and $b\in M_2\setminus\{2\}$.

MIPT student olimpiad spring 2023, 3

Tags: geometry , 3D geometry , sphere , college contests , MIPT

Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

2020-2021 OMMC, 2

Tags: ommc

There are a family of $5$ siblings. They have a pile of at least $2$ candies and are trying to split them up amongst themselves. If the $2$ oldest siblings share the candy equally, they will have $1$ piece of candy left over. If the $3$ oldest siblings share the candy equally, they will also have $1$ piece of candy left over. If all $5$ siblings share the candy equally, they will also have $1$ piece left over. What is the minimum amount of candy required for this to be true?

1996 All-Russian Olympiad Regional Round, 11.7

Tags: geometry , angles , similar triangles

In triangle $ABC$, a point $O$ is taken such that $\angle COA = \angle B + 60^o$, $\angle COB = \angle A + 60^o$, $\angle AOB = \angle C + 60^o$.Prove that if a triangle can be formed from the segments $AO$, $BO$, $CO$, then a triangle can also be formed from the altitudes of triangle $ABC$ and these triangles are similar.

1950 Miklós Schweitzer, 10

Tags: calculus , derivative , advanced fields , advanced fields unsolved

Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.

2005 Bosnia and Herzegovina Junior BMO TST, 3

Tags: Sequence , algebra

Rational numbers are written in the following sequence: $\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}, . . .$ In which position of this sequence is $\frac{2005}{2004}$ ?

Kvant 2019, M2556

Tags: RMM , RMM 2019 , number theory

Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win? [i]Maxim Didin, Russia[/i]

2012 Iran MO (3rd Round), 2

Tags: inequalities , geometry , combinatorics proposed , combinatorics

Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$. [i]Proposed by Ali Khezeli[/i]

1975 AMC 12/AHSME, 15

Tags: modular arithmetic

In the sequence of numbers 1, 3, 2, ... each term after the first two is equal to the term preceding it minus the term preceding that. The sum of the first one hundred terms of the sequence is $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \minus{}1$

2010 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry , 3D geometry , pyramid

$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$

LMT Team Rounds 2021+, 7

Tags: number theory

How many $2$-digit factors does $555555$ have?

2001 Mongolian Mathematical Olympiad, Problem 3

Tags: number theory , Divisibility

Let $a,b$ be coprime positive integers with $a$ even and $a>b$. Show that there exist infinitely many pairs $(m,n)$ of coprime positive integers such that $m\mid a^{n-1}-b^{n-1}$ and $n\mid a^{m-1}-b^{m-1}$.

2018 JBMO Shortlist, A3

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

2012 Romania Team Selection Test, 2

Tags: geometry , circumcircle , trigonometry , symmetry , geometric transformation , reflection , rectangle

Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.

2002 Iran MO (2nd round), 4

Tags: geometry , circumcircle , geometry unsolved

Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$

2008 Thailand Mathematical Olympiad, 3

Tags: floor function , algebra

Find all positive real solutions to the equation $x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor$

Durer Math Competition CD 1st Round - geometry, 2013.D3

Tags: geometry , ratio , areas , Durer

The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$? [img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]

2023 CCA Math Bonanza, TB2

Tags: CCA Math Bonanza , geometry , 3D geometry , tetrahedron

How many ways are there to color a tetrahedron’s faces, edges, and vertices in red, green, and blue so that no face shares a color with any of its edges, and no edge shares a color with any of its endpoints? (Rotations and reflections are considered distinct.) [i]Tiebreaker #2[/i]

2018 Korea National Olympiad, 6

Tags: combinatorics , number theory , algebra

Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition. For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.

2019 AMC 12/AHSME, 25

Tags: 2019 AMC 12B , AMC , AMC 12 , AMC 12 B

Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? $\textbf{(A) } 27 \qquad\textbf{(B) } 16\sqrt3 \qquad\textbf{(C) } 12+10\sqrt3 \qquad\textbf{(D) } 9+12\sqrt3 \qquad\textbf{(E) } 30$