Olympiad problems

Durer Math Competition CD Finals - geometry, 2023.C3

$ABC$ is an isosceles triangle. The base $BC$ is $1$ cm long, and legs $AB$ and $AC$ are $2$ cm long. Let the midpoint of $AB$ be $F$, and the midpoint of $AC$ be $G$. Additionally, $k$ is a circle, that is tangent to $AB$ and A$C$, and it’s points of tangency are $F$ and $G$ accordingly. Prove, that the intersection of $CF$ and $BG$ falls on the circle $k$.

2007 Balkan MO Shortlist, G2

Tags: trigonometry , geometry , geometric transformation , reflection , trapezoid , circumcircle , trig identities

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

1963 All Russian Mathematical Olympiad, 029

Tags: geometry , parallelogram , area

a) Each diagonal of the quadrangle halves its area. Prove that it is a parallelogram. b) Three main diagonals of the hexagon halve its area. Prove that they intersect in one point.

2021 AMC 12/AHSME Spring, 5

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When a student multiplied the number $66$ by the repeating decimal, $$1. \underline{a} \underline{b} \underline{a} \underline{b} … = 1.\overline{ab},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1. \underline{a} \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$- digit integer $\underline{a} \underline{b}$? $\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

Gheorghe Țițeica 2024, P2

Tags: algebra

Let $a,b,c>1$. Solve in $\mathbb{R}$ the equation $\log_{a+b}(a^x+b)=\log_b((b+c)^x-c)$. [i]Mihai Opincariu[/i]

2010 Slovenia National Olympiad, 5

Tags: geometry

For what positive integers $n \geq 3$ does there exist a polygon with $n$ vertices (not necessarily convex) with property that each of its sides is parallel to another one of its sides?

2009 Harvard-MIT Mathematics Tournament, 7

Tags: function

Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]

1953 AMC 12/AHSME, 30

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A house worth $ \$9000$ is sold by Mr. A to Mr. B at a $ 10\%$ loss. Mr. B sells the house back to Mr. A at a $ 10\%$ gain. The result of the two transactions is: $ \textbf{(A)}\ \text{Mr. A breaks even} \qquad\textbf{(B)}\ \text{Mr. B gains }\$900 \qquad\textbf{(C)}\ \text{Mr. A loses }\$900\\ \textbf{(D)}\ \text{Mr. A loses }\$810 \qquad\textbf{(E)}\ \text{Mr. B gains }\$1710$

LMT Accuracy Rounds, 2023 S6

Tags: combinatorics

Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?

1992 Tournament Of Towns, (336) 4

Tags: combinatorics , combinatorial geometry , area , geometry

Three triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$ are given such that their centres of gravity (intersection points of their medians) lie on a straight line, but no three of the $9$ vertices of the triangles lie on a straight line. Consider the set of $27$ triangles $A_iB_jC_k$ (where $i$, $j$, $k$ take the values $1$, $2$, $3$ independently). Prove that this set of triangles can be divided into two parts of the same total area. (A. Andjans, Riga)

2004 Turkey MO (2nd round), 1

Tags: symmedian , geometry , parallelogram , trapezoid , rectangle

In a triangle $\triangle ABC$ with$\angle B>\angle C$, the altitude, the angle bisector, and the median from $A$ intersect $BC$ at $H, L$ and $D$, respectively. Show that $\angle HAL=\angle DAL$ if and only if $\angle BAC=90^{\circ}$.

2016 Azerbaijan Team Selection Test, 2

Tags: algebra , polynomial

A positive interger $n$ is called [i][u]rising[/u][/i] if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called [i][u]interger-valued[/u][/i] if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called [i][u]rising-valued[/u][/i] if for all [i]rising[/i] numbers $n$, $P(n)$ takes integer values. Does it necessarily mean that, "every [i]rising-valued[/i] $P$ is also [i]interger-valued[/i] $P$"?

2014 Harvard-MIT Mathematics Tournament, 24

Tags:

Let $A=\{a_1,a_2,\ldots,a_7\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.

2018 Mathematical Talent Reward Programme, MCQ: P 5

Tags: trigonometry , algebra

Let the maximum and minimum value of $f(x)=\cos \left(x^{2018}\right) \sin x$ are $M$ and $m$ respectively where $x \in[-2 \pi, 2 \pi] .$ Then $$ M+m= $$ [list=1] [*] $\frac{1}{2}$ [*] $-\frac{1}{\sqrt{2}}$ [*] $\frac{1}{2018}$ [*] Does not exists [/list]

2016 May Olympiad, 5

Tags: game , combinatorics , geometry , winning strategy , area

Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

2011 Germany Team Selection Test, 2

Tags: floor function , number theory

Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$

2025 Harvard-MIT Mathematics Tournament, 4

Tags: guts

Let $\triangle{ABC}$ be an equilateral triangle with side length $4.$ Across all points $P$ inside triangle $\triangle{ABC}$ satisfying $[PAB]+[PAC]=[PBC],$ compute the minimal possible length of $PA.$ (Here, $[XYZ]$ denotes the area of triangle $\triangle{XYZ}.$)

2021 Argentina National Olympiad, 4

Tags: combinatorics

The sum of several positive integers, not necessarily different, all of them less than or equal to $10$, is equal to $S$. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to $80.$ Determine all values of $S$ for which this is possible.

2005 ISI B.Math Entrance Exam, 7

Tags: geometry , locus , area of a triangle

Let $M$ be a point in the triangle $ABC$ such that \[\text{area}(ABM)=2 \cdot \text{area}(ACM)\] Show that the locus of all such points is a straight line.

1994 Hong Kong TST, 2

Tags: function , modular arithmetic , algebra

Given that, a function $f(n)$, defined on the natural numbers, satisfies the following conditions: (i)$f(n)=n-12$ if $n>2000$; (ii)$f(n)=f(f(n+16))$ if $n \leq 2000$. (a) Find $f(n)$. (b) Find all solutions to $f(n)=n$.

2023 JBMO Shortlist, N1

Tags: number theory

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

1999 Harvard-MIT Mathematics Tournament, 1

Tags: limit

Start with an angle of $60^\circ$ and bisect it, then bisect the lower $30^\circ$ angle, then the upper $15^\circ$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^\circ$ angle into two angles. Find the measure (degrees) of the smaller angle.

2002 China Team Selection Test, 3

Tags: inequalities

$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and: \[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\] and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.

2023 MOAA, 10

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Let $S$ be a set of integers such that if $a$ and $b$ are in $S$ then $3a-2b$ is also in $S$. How many ways are there to construct $S$ such that $S$ contains exactly $4$ elements in the interval $[0,40]$? [i]Proposed by Harry Kim[/i]

IV Soros Olympiad 1997 - 98 (Russia), 9.7

Tags: algebra , percentage

There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5: 4: 3,$ you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?