Olympiad problems

2004 Estonia National Olympiad, 3

On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

Cono Sur Shortlist - geometry, 2009.G4

Tags: geometry , perpendicular , altitudes , incenter

Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.

1999 All-Russian Olympiad Regional Round, 8.1

Tags: algebra , combinatorics

A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours

2021-2022 OMMC, 12

Tags: ommc

Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11$) is $\tfrac ab$ for coprime positive integers $a$, $b$. Find $a + b$. [i]Proposed by Evan Chang[/i]

2014 China Western Mathematical Olympiad, 7

Tags: inequalities proposed , inequalities

In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$. Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]

2004 Romania National Olympiad, 2

Tags: algebra , polynomial , function , superior algebra , superior algebra solved

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

Today's calculation of integrals, 869

Tags: calculus , integration , trigonometry , calculus computations

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

2017 AMC 12/AHSME, 9

Tags: AMC , AMC 12 , AMC 12 A , 2017 AMC 12A , AMC 10 , 2017 AMC 10A , Sets

Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$? $\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$

1953 AMC 12/AHSME, 31

Tags:

The rails on a railroad are $ 30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in miles per hour is approximately the number of clicks heard in: $ \textbf{(A)}\ 20\text{ seconds} \qquad\textbf{(B)}\ 2\text{ minutes} \qquad\textbf{(C)}\ 1\frac{1}{2}\text{ minutes} \qquad\textbf{(D)}\ 5\text{ minutes}\\ \textbf{(E)}\ \text{none of these}$

2014-2015 SDML (High School), 1

Tags: 2014-15 SDML Round 1 , SDML High School , SDML Middle School

If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?

2002 Estonia Team Selection Test, 1

Tags: number theory , combinatorics

The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.

2016 AIME Problems, 11

Tags: AMC , AIME , AIME II

For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.

2010 Putnam, A5

Tags: Putnam , vector , college contests

Let $G$ be a group, with operation $*$. Suppose that (i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors); (ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$ Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$

2018 ELMO Shortlist, 3

Tags: inequalities

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\] [i]Proposed by Daniel Liu[/i]

2015 Chile TST Ibero, 4

Tags: inequalities , TST , Chile , algebra

Let $x, y \in \mathbb{R}^+$. Prove that: \[ \left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2. \]

2021 239 Open Mathematical Olympiad, 4

Tags: algebra , inequalities

Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.

2013 Online Math Open Problems, 8

Tags: Online Math Open

How many ways are there to choose (not necessarily distinct) integers $a,b,c$ from the set $\{1,2,3,4\}$ such that $a^{(b^c)}$ is divisible by $4$? [i]Ray Li[/i]

2025 Romania National Olympiad, 2

Tags: combinatorial geometry , minimum , combinatorics

Let $n$ be a positive integer. For a set of points in the plane $M$, we call $2$ distinct points $A,B \in M$ [i]connected[/i] if the line $AB$ contains exactly $n+1$ points from $M$. Find the minimum value of a positive integer $m$ such that there exists a set $M$ of $m$ points in the plane with the property that any point $A \in M$ is connected with exactly $2n$ other points from $M$.

2016 Iranian Geometry Olympiad, 2

Tags: geometry , equal segments , circumcircle , Angle Chasing

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$. by Iman Maghsoudi

2018 Saudi Arabia GMO TST, 4

Tags: combinatorics , square board

In each of the cells of a $13 \times 13$ board is written an integer such that the integers in adjacent cells differ by $1$. If there are two $2$s and two $24$s on this board, how many $13$s can there be?

1998 Romania National Olympiad, 1

Tags: algebra , system of equations , inequalities

Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .

2016 Romanian Masters in Mathematic, 4

Tags: algebra , inequalities , RMM , RMM 2016

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

2025 CMIMC Algebra/NT, 4

Tags: algebra , number theory

Consider the system of equations $$\log_x y +\log_y z + \log_z x =8$$ $$\log_{\log_y x}z = -3$$ $$\log_z y + \log_x z = 16$$ Find $z.$

1991 Romania Team Selection Test, 2

Tags: concurrent planes , concurrent , planes , tetrahedron

Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote: - $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$; - $B_{ij}$ – the midpoint of the edge $A_iAj$, - $C_{ij}$ – the midpoint of segment $P_iP_j$ - $\beta_{ij}$– the plane $B_{ij}P_hP_k$ - $\delta_{ij}$ – the plane $B_{ij}P_iP_j$ - $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$ - $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$. Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent: (a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.

2007 Tournament Of Towns, 5

Tags: geometry , 3D geometry , pyramid , octahedron

From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?