Olympiad problems

STEMS 2023 Math Cat A, 3

Tags: geometry , ratio

Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$, let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.

2018 Turkey Team Selection Test, 5

Tags: combinatorics

We say that a group of $25$ students is a [i]team[/i] if any two students in this group are friends. It is known that in the school any student belongs to at least one team but if any two students end their friendships at least one student does not belong to any team. We say that a team is [i]special[/i] if at least one student of the team has no friend outside of this team. Show that any two friends belong to some special team.

VI Soros Olympiad 1999 - 2000 (Russia), 8.5

Tags: number theory , system of equations

Solve the following system of equations in natural numbers $$\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}$$

2010 Princeton University Math Competition, 8

Tags: geometry , 3d geometry , sphere , tetrahedron

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

1949-56 Chisinau City MO, 1

Tags: algebra

The numbers $1, 2, ..., 1000$ are written out in a row along a circle. Starting from the first, every fifteenth number in the circle is crossed out $(1, 16, 31, ...)$, in this case, the crossed out numbers are still taken into account at each new round of the circle. How many numbers are left uncrossed?

1938 Moscow Mathematical Olympiad, 041

Tags: geometric construction , triangle construction , geometry

Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.

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? $\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$