Olympiad problems

1994 Spain Mathematical Olympiad, 1

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

2020 Macedonian Nationаl Olympiad, 3

Tags: geometry , Ugly , Angle Bisectors , angle bisector , circumcircle

Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.

2020 HMNT (HMMO), 1

Tags: geometry , analytic geometry

In the Cartesian plane, a line segment with midpoint $(2020,11)$ has one endpoint at $(a,0)$ and the other endpoint on the line $y=x$. Compute $a$.

2023 Bulgarian Spring Mathematical Competition, 12.3

Tags: factorial , number theory

Given is a polynomial $f$ of degree $m$ with integer coefficients and positive leading coefficient. A positive integer $n$ is $\textit {good for f(x)}$ if there exists a positive integer $k_n$, such that $n!+1=f(n)^{k_n}$. Prove that there exist only finitely many integers good for $f$.

2018 Malaysia National Olympiad, B2

Tags: Sets , Proof , number theory

A subset of $\{1, 2, 3, ... ... , 2015\}$ is called good if the following condition is fulfilled: for any element $x$ of the subset, the sum of all the other elements in the subset has the same last digit as $x$. For example, $\{10, 20, 30\}$ is a good subset since $10$ has the same last digit as $20 + 30 = 50$, $20$ has the same last digit as $10 + 30 = 40$, and $30$ has the same last digit as $10 + 20 = 30$. (a) Find an example of a good subset with 400 elements. (b) Prove that there is no good subset with 405 elements.

2022 CCA Math Bonanza, I1

Tags:

Asteroids A and B have circular orbits around the same star. Asteroid A is located 400 km away from the star and takes 8000 hours to complete one full revolution. Asteroid B is located 100 km away and the speed of Asteroid B is twice the speed of Asteroid A. Find how long it takes for Asteroid B to complete one full revolution in hours. [i]2022 CCA Math Bonanza Individual Round #1[/i]

Kyiv City MO 1984-93 - geometry, 1985.8.3

Tags: geometry , geometric inequality

The longest diagonal of a convex hexagon is $2$. Is there necessarily a side or diagonal in this hexagon whose length does not exceed $1$?

2007 AMC 10, 10

Tags: geometry , AMC , AMC 10

The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $ 20$, the father is $ 48$ years old, and the average age of the mother and children is $ 16$. How many children are in the family? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

2022 IMO Shortlist, G7

Tags: Euler , geometry , circumcircle

Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.

2008 Sharygin Geometry Olympiad, 11

Tags: geometry proposed , geometry

(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

1996 Moscow Mathematical Olympiad, 1

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It is known that $a+\frac{b^2}{a}=b+\frac{a^2}{b}$. Is it true that $a=b$, where $a$ and $b$ are nonzero real numbers? Proposed by R.Fedorov

2002 AMC 8, 25

Tags:

Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have? $\text{(A)}\ \frac{1}{10} \qquad \text{(B)}\ \frac{1}{4} \qquad \text{(C)}\ \frac{1}{3} \qquad \text{(D)}\ \frac{2}{5} \qquad \text{(E)}\ \frac{1}{2}$

1966 IMO Shortlist, 30

Tags: inequalities , logarithms , calculus , IMO Shortlist , IMO Longlist

Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2023 South East Mathematical Olympiad, 7

Tags: algebra , inequalities

The positive integer number $S$ is called a "[i]line number[/i]". if there is a positive integer $n$ and $2n$ positive integers $a_1$, $a_2$,...,$a_n$, $b_1$,$b_2$,...,$b_n$, such that $S = \sum^n_{i=1} a_ib_i$, $\sum^n_{i=1} (a_i^2-b_1^2)=1$, and $\sum^n_{i=1} (a_i+b_i)=2023$, find: (1) The minimum value of [i]line numbers[/i]. (2)The maximum value of [i]line numbers[/i].

2005 Denmark MO - Mohr Contest, 1

Tags: geometry , 3D geometry , pyramid , area

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

May Olympiad L1 - geometry, 2019.4

Tags: geometry , combinatorics , square , cutting the paper , paper , Obtuse triangle

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

2015 JBMO Shortlist, A3

Tags: Zhan , SBYT , QJYT

If $a,b,c$ are positive real numbers prove that: $\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}>2.$

2024 CMIMC Team, 5

Tags: team

An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die. Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.) [i]Proposed by Lohith Tummala[/i]

2001 Estonia National Olympiad, 4

Tags: Composite , Sum of powers , prime , number theory

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

2023 Romania National Olympiad, 2

Tags: algebra , trigonometry

Determine the largest natural number $k$ such that there exists a natural number $n$ satisfying: \[ \sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k). \]

2010 Today's Calculation Of Integral, 613

Tags: calculus , integration , geometry , logarithms , calculus computations

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

1999 IMC, 4

Tags: function , real analysis , real analysis unsolved

Find all strictly monotonic functions $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ for which $f\left(\frac{x^2}{f(x)}\right)=x$ for all $x$.

2015 Thailand Mathematical Olympiad, 1

Tags: divides , recurrence relation , divisible , number theory

Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

1998 Argentina National Olympiad, 6

Tags: inequalities , algebra

Given $n$ non-negative real numbers, $n\geq 3$, such that the sum of the $n$ numbers is less than or equal to $3$ and the sum of the squares of the $n$ numbers is greater than or equal to $1$, prove that among the $n$ numbers three can be chosen whose sum is greater than or equal to $1$.

2014 India PRMO, 8

Tags: Sets , arithmetic mean

Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have?