Olympiad problems

2020 Yasinsky Geometry Olympiad, 4

The median $AM$ is drawn in the triangle $ABC$ ($AB \ne AC$). The point $P$ is the foot of the perpendicular drawn on the segment $AM$ from the point $B$. On the segment $AM$ we chose such a point $Q$ that $AQ = 2PM$. Prove that $\angle CQM = \angle BAM$.

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: diophantine , number theory , radical , inequalities

Let $n$ be a positive integer. Prove that there are no positive integers $x$ and $y$ such as $\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2} $

1979 IMO Longlists, 9

Tags: inequalities , function , logarithm

The real numbers $\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n$ are positive. Let us denote by $h = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n}$ the harmonic mean, $g=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n}$ the geometric mean, and $a=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n}$ the arithmetic mean. Prove that $h \leq g \leq a$, and that each of the equalities implies the other one.

1993 Mexico National Olympiad, 2

Tags: number theory , sum of cubes

Find all numbers between $100$ and $999$ which equal the sum of the cubes of their digits.

2018 Iran MO (1st Round), 19

Tags: inequalities , algebra

Let $x \geq y \geq z$ be positive real numbers such that \begin{align*}x^2+y^2+z^2 \geq 2xy+2yz+2zx.\end{align*} What is the minimum value of $\frac{x}{z}$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt 2\qquad\textbf{(C)}\ \sqrt 3\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

2024 MMATHS, 8

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Triangle $ABC$ is an acute triangle with $BC=6$ and $AC=7.$ Let $D, E,$ and $F$ be the feet of the altitudes from $A, B,$ and $C$ respectively. $\overline{AD}$ bisects angle $FDE.$ Let $m$ be the maximum possible value of $FD+ED.$ Find $m^2.$

2021 AMC 12/AHSME Fall, 1

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What is the value of $\frac{(2112-2021)^2}{169}$? $\textbf{(A) }7\qquad\textbf{(B) }21\qquad\textbf{(C) }49\qquad\textbf{(D) }64\qquad\textbf{(E) }91$

2021 Romania National Olympiad, 3

Tags: geometry

Let $ABC$ be a scalene triangle with $\angle BAC>90^\circ$. Let $D$ and $E$ be two points on the side $BC$ such that $\angle BAD=\angle ACB$ and $\angle CAE=\angle ABC$. The angle-bisector of $\angle ACB$ meets $AD$ at $N$, If $MN\parallel BC$, determine $\angle (BM, CN)$. [i]Petru Braica[/i]

2016 PAMO, 3

Tags: number theory , integer polynomial , greatest common divisor

For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.

1953 Moscow Mathematical Olympiad, 240

Tags: midpoint , skew , geometric inequality , 3d geometry , geometry

Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.

1998 Estonia National Olympiad, 4

Tags: divide , number theory , divisible , prime

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

1982 Putnam, B1

Tags: geometry

Let $M$ be the midpoint of side $BC$ of $\triangle ABC$. Using the [i]smallest possible[/i] $n$, described a method for cutting $\triangle AMB$ into $n$ triangles which can be reassembled to form a triangle congruent to $\triangle AMC$.

2018 AMC 10, 18

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How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? $\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048 $

2010 Contests, 2

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Denote by $S(n)$ the sum of digits of integer $n.$ Find 1) $S(3)+S(6)+S(9)+\ldots+S(300);$ 2) $S(3)+S(6)+S(9)+\ldots+S(3000).$

2021 Indonesia TST, A

Tags: function , algebra , functional inequality

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

2007 AMC 12/AHSME, 22

Tags: ratio , vector , function , geometry

Two particles move along the edges of equilateral triangle $ \triangle ABC$ in the direction \[ A\rightarrow B\rightarrow C\rightarrow A \]starting simultaneously and moving at the same speed. One starts at $ A$, and the other starts at the midpoint of $ \overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $ R$. What is the ratio of the area of $ R$ to the area of $ \triangle ABC$? $ \textbf{(A)}\ \frac {1}{16}\qquad \textbf{(B)}\ \frac {1}{12}\qquad \textbf{(C)}\ \frac {1}{9}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$

2010 Indonesia TST, 2

Tags: greatest common divisor , number theory , sequence

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

1999 Chile National Olympiad, 7

Tags: algebra , functional equation , functional

Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$. $\bullet$ $f (1998) = 2$ Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,

2009 Indonesia TST, 4

Tags: geometry

Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: perimeter , geometry

A non-equilateral triangle $\triangle ABC$ of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$. It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the segment $BC$. [i] (А. Кузнецов)[/i]

1990 Greece National Olympiad, 4

Tags: number theory , last digit , digit

Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

1940 Eotvos Mathematical Competition, 1

Tags: combinatorics

In a set of objects, each has one of two colors and one of two shapes. There is at least one object of each color and at least one object of each shape. Prove that there exist two objects in the set that are different both in color and in shape.

2002 National High School Mathematics League, 2

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For real numbers $a,b,c$ and positive number $\lambda$ such that three real roots $x_1,x_2,x_3$ of $f(x)=x^3+ax^2+bx+c$ satisfying: $(1) x_2-x_1=\lambda$; $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$

1996 Tournament Of Towns, (523) 6

Tags: combinatorics

The integers from $1$ to $100$ are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose $10 $ of these $100$ numbers. Then 10 of the integers from $1$ to $100$ are drawn, and a winning ticket is one which does not contain any of them. Prove that (a) if you buy $13$ tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket; (b) if you buy only $12$ tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers. (S Tokarev)

2015 China Western Mathematical Olympiad, 2

Tags: angle bisector , geometry

Two circles $ \left(\Omega_1\right),\left(\Omega_2\right) $ touch internally on the point $ T $. Let $ M,N $ be two points on the circle $ \left(\Omega_1\right) $ which are different from $ T $ and $ A,B,C,D $ be four points on $ \left(\Omega_2\right) $ such that the chords $ AB, CD $ pass through $ M,N $, respectively. Prove that if $ AC,BD,MN $ have a common point $ K $, then $ TK $ is the angle bisector of $ \angle MTN $. * $ \left(\Omega_2\right) $ is bigger than $ \left(\Omega_1\right) $