Olympiad problems

2021 Argentina National Olympiad Level 2, 2

Tags: geometry

In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$

2011 Croatia Team Selection Test, 2

Tags: analytic geometry , combinatorics

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

2015 ASDAN Math Tournament, 9

Tags: algebra test

Compute all pairs of nonzero real numbers $(x,y)$ such that $$\frac{x}{x^2+y}+\frac{y}{x+y^2}=-1\qquad\text{and}\qquad\frac{1}{x}+\frac{1}{y}=1.$$

2021 USMCA, 18

Tags:

Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integer(s). He first rolls the die and if the number showing on top is $k,$ he then uniformly and randomly takes a $k$-element subset from his list and calculates the sum of the numbers in his subset. Given that the expected value of this sum is $2020,$ compute the sum of all possible values of $n.$

1996 Estonia National Olympiad, 4

Tags: number theory , prime number

Can the remainder of the division of a prime number $p> 30$ by $30$ be a composite?

1996 Greece Junior Math Olympiad, 1

Tags: equation , algebra

Solve the equation $(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$

2021 CHMMC Winter (2021-22), 1

Tags: geometry

Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$, point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $\overline{OE}$, respectively. Compute the area of quadrilateral $W XY Z$.

1941 Moscow Mathematical Olympiad, 085

Tags: prime , remainder , number theory

Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.

1995 Romania Team Selection Test, 3

Tags: geometry , geometric transformation , rotation , combinatorics

Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.

1973 Bundeswettbewerb Mathematik, 4

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$n$ persons sit around a round table. The number of persons having the same gender than the person at the right of them is the same as the number of those it isn't true for. Show that $4|n$.

2020 Ukrainian Geometry Olympiad - April, 3

Tags: geometry , parallelogram , orthocenter , angle inequalities , angle

Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.

2017 District Olympiad, 3

Tags: arithmetic sequence , number theory

Denote $ S_n $ as being the sum of the squares of the first $ n\in\mathbb{N} $ terms of a given arithmetic sequence of natural numbers. [b]a)[/b] If $ p\ge 5 $ is a prime, then $ p\big| S_p. $ [b]b)[/b] $ S_5 $ is not a perfect square.

2017 Romania National Olympiad, 1

Tags: real analysis , function , continuity , sequence

Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.

2015 Korea National Olympiad, 4

Tags: inequalities , number theory , relatively prime , prime number , coprime integers

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

1971 Miklós Schweitzer, 8

Tags: combinatorics

Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $ e$ there is a simple directed path from pole $ p$ to pole $ q$ containing $ e$. (A strongly connected bipolar graph is a finite connected graph with two special vertices $ p$ and $ q$ having the property that there are no points $ x,y,x \not \equal{} y$, such that all paths from $ x$ to $ p$ as well as all paths from $ x$ to $ q$ contain $ y$.) [i]A. Adam[/i]

1951 Poland - Second Round, 2

Tags: geometry , inscribed , ratio , area

In the triangle $ ABC $ on the sides $ BC $, $ CA $, $ AB $, the points $ D $, $ E $, $ F $ are chosen respectively in such a way that $$ BD \colon DC = CE \colon EA = AF \colon FB = k,$$ where $k$ is a given positive number. Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle $ DEF $

2009 Harvard-MIT Mathematics Tournament, 6

Tags:

How many sequences of $5$ positive integers $(a,b,c,d,e)$ satisfy $abcde\leq a+b+c+d+e\leq10$?

2012 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: number theory

Given a positive integer $n$, let $d(n)$ be the largest positive divisor of $n$ less than $n$. For example, $d(8) = 4$ and $d(13) = 1$. A sequence of positive integers $a_1, a_2,\dots$ satisfies \[a_{i+1} = a_i +d(a_i),\] for all positive integers $i$. Prove that regardless of the choice of $a_1$, there are inﬁnitely many terms in the sequence divisible by $3^{2011}$.

2016 AMC 10, 6

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Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's? $\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$

1990 AMC 12/AHSME, 28

Tags:

A quadrilateral that has consecutive sides of lengths $70, 90, 130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$. $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16 $

2002 All-Russian Olympiad, 4

Tags: combinatorics

On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.

2010 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).

2018 CCA Math Bonanza, TB3

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Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$? [i]2018 CCA Math Bonanza Tiebreaker Round #3[/i]

2024 USAJMO, 1

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral. [i]Proposed by Evan O'Dorney[/i]

2006 Korea National Olympiad, 6

Tags: inequalities , algebra

Prove that for any positive real numbers $x,y$ and $z,$ $xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$