Olympiad problems

2001 Tournament Of Towns, 1

A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)

2018 Iran MO (1st Round), 10

Tags: geometry

Consider a triangle $ABC$ in which $AB=AC=15$ and $BC=18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD=5$ and $CE=3$. The point $F$ is chosen on the half-line $\overrightarrow{DE}$ so that $EF=8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what is the length of $CN$?

1957 Miklós Schweitzer, 10

Tags: abstract algebra , group theory , college contests

[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]

2010 IFYM, Sozopol, 2

Tags: combinatorics , table , Coloring

Is it possible to color the cells of a table 19 x 19 in yellow, blue, red, and green so that each rectangle $a$ x $b$ ($a,b\geq 2$) in the table has at least 2 cells in different color?

2017 QEDMO 15th, 11

Tags: algebra , functional , functional equation

Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.

1999 USAMTS Problems, 4

Tags: USAMTS , analytic geometry , calculus , integration , geometry , perimeter

We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths. (a) Find an integral triangle with perimeter of $42$. (b) Is there an integral triangle with perimeter of $43$?

2017 Purple Comet Problems, 3

Tags: Purple Comet

The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?

2010 Regional Competition For Advanced Students, 2

Tags: algebra proposed , algebra

Solve the following in equation in $\mathbb{R}^3$: \[4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.\]

2012 Purple Comet Problems, 4

Tags: complementary counting

How many two-digit positive integers contain at least one digit equal to 5?

2001 Romania National Olympiad, 1

Tags: function , calculus , derivative , real analysis , real analysis unsolved

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that: \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]

2015 USA Team Selection Test, 2

Tags: graph theory , combinatorics , Coloring , Tournament , Directed graphs , Hi

A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors. Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices. [i]Proposed by Po-Shen Loh[/i]

2010 Vietnam Team Selection Test, 3

Tags: algebra , polynomial , modular arithmetic , number theory unsolved , number theory

Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$

2018 AMC 10, 2

Tags: AMC , AMC 10 , AMC 10 A

Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have? $ \textbf{(A) }\text{ Liliane has } 20\%\text{ more soda than Alice.}$ $\textbf{(B) }\text{ Liliane has } 25\%\text{ more soda than Alice.}$ $\textbf{(C) }\text{ Liliane has } 45\%\text{ more soda than Alice.}$ $ \textbf{(D) }\text{ Liliane has } 75\%\text{ more soda than Alice.}$ $\textbf{(E) }\text{ Liliane has } 100\%\text{ more soda than Alice.}$

2008 District Olympiad, 2

Tags: linear algebra , matrix , linear algebra unsolved

Let $A,B\in \mathcal{M}_n(\mathbb{R})$. Prove that $\text{rank}\ A+\text{rank}\ B\le n$ if and only if there exists an invertible matrix $X\in \mathcal{M}_n(\mathbb{R})$ such that $AXB=O_n$.

2020/2021 Tournament of Towns, P4

Tags: combinatorics , game , Tournament of Towns

There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays? [i]Ivan Mitrofanov[/i]

2024 PErA, P3

Tags: inequalities

Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$ where we denote $x_0=x_n$ and $x_{n+1}=x_1$.

1997 AMC 12/AHSME, 13

Tags: calculus , integration

How many two-digit positive integers $ N$ have the property that the sum of $ N$ and the number obtained by reversing the order of the digits of $ N$ is a perfect square? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

2018 CCA Math Bonanza, L1.4

Tags:

What is the sum of all distinct values of $x$ that satisfy $x^4-x^3-7x^2+13x-6=0$? [i]2018 CCA Math Bonanza Lightning Round #1.4[/i]

2003 Tournament Of Towns, 5

Tags: geometry , 3D geometry , tetrahedron , combinatorics unsolved , combinatorics

A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?

1987 Tournament Of Towns, (158) 2

Tags: time , combinatorics , speed , game , game strategy

In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?

2016 South East Mathematical Olympiad, 2

Tags: inequalities

Let $n$ be positive integer,$x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1x_2\cdots x_n=1 $ . Prove that$$\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}$$

2012 China Western Mathematical Olympiad, 3

Tags: combinatorics proposed , combinatorics

Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$

2014-2015 SDML (High School), 2

Tags: SDML High School , 2014-15 SDML Round 2 , Divisibility , SDML Middle School

Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?

2010 Romania Team Selection Test, 2

Tags: function , parameterization , induction , inequalities , rearrangement inequality , algebra proposed , algebra

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

2006 Balkan MO, 2

Tags: geometry , circumcircle , projective geometry , algebra , polynomial , power of a point , radical axis

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point. [i]Greece[/i]