Olympiad problems

2004 China Team Selection Test, 3

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

2006 Romania National Olympiad, 4

Tags: real analysis , derivative , calculus , integration , trigonometry , function

Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \] [i]Cezar Lupu, Tudorel Lupu[/i]

2020 AMC 8 -, 23

Tags:

Five different awards are to be given to three students. Each student will receive at least one award. In how many ways can the awards be distributed? $\textbf{(A)}\ 120 \qquad \textbf{(B)}\ 150 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 210 \qquad \textbf{(E)}\ 240$

2010 Saudi Arabia IMO TST, 3

Tags: functional , function , functional equation , algebra

Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$. Note: $N = \{0,1,2,...\}$

1957 Putnam, A7

Tags: tangency , circles

Each member of a set of circles in the $xy$-plane is tangent to the $x$-axis and no two of the circles intersect. Show that (a) the points of tangency can include all rational points on the axis. (b) the points of tangency cannot include all the irrational points.

2008 JBMO Shortlist, 6

Tags: inequalities , algebra

If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.

1995 Israel Mathematical Olympiad, 5

Tags: inequalities , algebra

Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$. Prove that $$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$

2008 JBMO Shortlist, 4

Tags: coloring , combinatorics

Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

2014 Iran MO (3rd Round), 4

Tags: number theory

$2 \leq d$ is a natural number. $B_{a,b}$={$a,a+b,a+2b,...,a+db$} $A_{c,q}$={$cq^n \vert n \in\mathbb{N}$} Prove that there are finite prime numbers like $p$ such exists $a,b,c,q$ from natural numbers : $i$ ) $ p \nmid abcq $ $ ii$ ) $A_{c,q} \equiv B_{a,b} (mod p ) $ (15 points )

2016 BMT Spring, 15

Tags: algebra

Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$. $$\prod_{i=1}^{3}(4s^4_i + 81)$$ can be written as $2^a3^b5^c$. Find $a + b + c$.

2016 Purple Comet Problems, 29

Tags:

Ten square tiles are placed in a row, and each can be painted with one of the four colors red (R), yellow (Y), blue (B), and white (W). Find the number of ways this can be done so that each block of five adjacent tiles contains at least one tile of each color. That is, count the patterns RWBWYRRBWY and WWBYRWYBWR but not RWBYYBWWRY because the five adjacent tiles colored BYYBW does not include the color red.

2023 Hong Kong Team Selection Test, Problem 1

Tags: inequality , easy , algebra

Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

1995 Vietnam National Olympiad, 3

Tags: geometry

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

2007 Today's Calculation Of Integral, 187

Tags: trigonometry , integration , calculus , calculus computations , inequalities

For a constant $a,$ let $f(x)=ax\sin x+x+\frac{\pi}{2}.$ Find the range of $a$ such that $\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).$

Estonia Open Junior - geometry, 2014.1.5

Tags: midpoint , geometry , concurrency

In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.

1972 AMC 12/AHSME, 31

Tags: relatively prime , modular arithmetic , number theory

When the number $2^{1000}$ is divided by $13$, the remainder in the division is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }7\qquad \textbf{(E) }11$

2019 Singapore Junior Math Olympiad, 5

Tags: combinatorics

Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.

1994 IberoAmerican, 1

Tags: number theory

A number $n$ is said to be [i]nice[/i] if it exists an integer $r>0$ such that the expression of $n$ in base $r$ has all its digits equal. For example, 62 and 15 are $\emph{nice}$ because 62 is 222 in base 5, and 15 is 33 in base 4. Show that 1993 is not [i]nice[/i], but 1994 is.

1995 All-Russian Olympiad, 3

Tags: number theory , algorithm , modular arithmetic

Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$? [i]A. Shapovalov[/i]

2012 Brazil Team Selection Test, 1

Tags: algorithm , number theory , prime number , divisibility , number of divisors

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

2022 CCA Math Bonanza, L4.3

Tags:

Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing? [i]2022 CCA Math Bonanza Lightning Round 4.3[/i]

2020 Ukrainian Geometry Olympiad - December, 3

Tags: median , angle , geometry , isosceles

In a triangle $ABC$ with an angle $\angle CAB =30^o$ draw median $CD$. If the formed $\vartriangle ACD$ is isosceles, find tan $\angle DCB$.

2021 LMT Fall, 1

Tags: algebra

Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

1974 IMO Longlists, 10

Tags: geometry , combinatorics

A regular octagon $P$ is given whose incircle $k$ has diameter $1$. About $k$ is circumscribed a regular $16$-gon, which is also inscribed in $P$, cutting from $P$ eight isosceles triangles. To the octagon $P$, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$-gon so obtained is said to be $P'$. Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$, one of the $11$-gons $P'$ contains all of $M$.

2009 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry

Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$