Olympiad problems

1993 All-Russian Olympiad, 2

Tags: symmetry , geometry

From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle.

2020 Belarusian National Olympiad, 11.2

Tags: geometry

Let $I$ be the incenter of a triangle $ABC$ with the property $\angle ABC - \angle BAC=30^{\circ}$. Line $CI$ intersects the circumcircle of $ABC$ at $C_1$. It turned out that $C_1$ lies on a common tangent line of circumcircles of triangles $ABC$ and $BCI$. Find the angles of triangle $ABC$.

1988 IberoAmerican, 3

Tags: perimeter , ellipse , geometry , conic

Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.

1968 Spain Mathematical Olympiad, 8

Tags: geometry , geometric transformation , reflection

We will assume that the sides of a square are reflective and we will designate them with the names of the four cardinal points. Marking a point on the side $N$ , determine in which direction a ray of light should exit (into the interior of the square) so that it returns to it after having undergone $n$ reflections on the side $E$ , another $n$ on the side $W$ , $m$ on the $S$ and $m - 1$ on the $N$, where $n$ and $m$ are known natural numbers. What happens if m and $n$ are not prime to each other? Calculate the length of the light ray considered as a function of $m$ and $n$, and of the length of the side of the square.

KoMaL A Problems 2021/2022, A. 824

Tags: algebra

An infinite set $S$ of positive numbers is called thick, if in every interval of the form $\left [1/(n+1),1/n\right]$ (where $n$ is an arbitrary positive integer) there is a number which is the difference of two elements from $S$. Does there exist a thick set such that the sum of its elements is finite? Proposed by [i]Gábor Szűcs[/i], Szikszó

1990 Romania Team Selection Test, 4

Tags: hexahedron , 3d geometry , geometry , sphere

The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.

2008 ITest, 92

Tags:

Find [the decimal form of] the largest prime divisor of $100111011_6$.

2012 Tournament of Towns, 7

Tags: combinatorics

There are $1 000 000$ soldiers in a line. The sergeant splits the line into $100$ segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most $100$ of these numbers are different.

2018 Sharygin Geometry Olympiad, 5

Tags: radical axis , arc midpoint , geometry

Let $ABCD$ be a cyclic quadrilateral, $BL$ and $CN$ be the internal angle bisectors in triangles $ABD$ and $ACD$ respectively. The circumcircles of triangles $ABL$ and $CDN$ meet at points $P$ and $Q$. Prove that the line $PQ$ passes through the midpoint of the arc $AD$ not containing $B$.

VII Soros Olympiad 2000 - 01, 10.1

Tags: algebra

Find all values of the parameter $a$ for which the equation $$(a-1)^2x^4 + (a^2-a) x^3 + 3x - 1 = 0$$ has a unique solution and for these $a$ solve the equation.

2018 Sharygin Geometry Olympiad, 1

Tags: geometry

Three circles lie inside a square. Each of them touches externally two remaining circles. Also each circle touches two sides of the square. Prove that two of these circles are congruent.

Russian TST 2016, P4

Tags: combinatorics

A regular $n{}$-gon and a regular $m$-gon with distinct vertices are inscribed in the same circle. The vertices of these polygons divide the circle into $n+m$ arcs. Is it always possible to inscribe a regular $(n+m)$-gon in the same circle so that exactly one of its vertices is on each of these arcs?

2010 China Northern MO, 6

Tags: geometry , bisects segment , incircle

Let $\odot O$ be the inscribed circle of $\vartriangle ABC$, with $D$, $E$, $N$ the touchpoints with sides $AB$, $AC$, $BC$ respectively. Extension of $NO$ intersects segment $DE$ at point $K$. Extension of $AK$ intersects segment $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/a503c500178551ddf9bdb1df0805ed22bc417d.png[/img]

VI Soros Olympiad 1999 - 2000 (Russia), 9.8

Tags: algebra , trigonometry , sum

Let $a_n$ denote an angle from the interval for each $\left( 0, \frac{\pi}{2}\right)$ , the tangent of which is equal to $n$ . Prove that $$\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000} $$

2007 Germany Team Selection Test, 3

Tags: number theory

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

2005 Today's Calculation Of Integral, 46

Tags: calculus , integration , calculus computations , logarithm

Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$

2011 USA Team Selection Test, 8

Tags: induction , graph theory , combinatorics

Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.

2021 Olympic Revenge, 1

Tags: inequality , algebra , inequalities

Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that: \[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\] [i]Proposed by Zhang Yanzong and Song Qing[/i]

1966 German National Olympiad, 1

Tags: algebra , number theory

Determine all real numbers $a, b$ and all integers $n\ge 1$ for which$ (a + b)^n = a^n + b^n$ holds.

1999 AMC 12/AHSME, 29

Tags: geometry , 3d geometry , tetrahedron , sphere , probability , inradius , circumcircle

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 0.1\qquad \textbf{(C)}\ 0.2\qquad \textbf{(D)}\ 0.3\qquad \textbf{(E)}\ 0.4$

2020 SIME, 15

Tags:

Triangle $\triangle ABC$ has side lengths $\overline{AB} = 13, \overline{BC} = 14,$ and $\overline{AC} = 15$. Suppose $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $P$ be a point on $\overline{MN}$, such that if the circumcircles of triangles $\triangle BMP$ and $\triangle CNP$ intersect at a second point $Q$ distinct from $P$, then $PQ$ is parallel to $AB$. The value of $AP^2$ can be expressed as a common fraction of the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

II Soros Olympiad 1995 - 96 (Russia), 9.3

Tags: combinatorics , combinatorial geometry , geometry

It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.

2020 MIG, 13

Tags:

For how many real values of $x$ is the equation $(x^2 - 7)^3 = 0$ true? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

1997 Turkey Team Selection Test, 2

Tags: modular arithmetic , abstract algebra , number theory

Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$

2005 Tournament of Towns, 4

Tags: ratio , geometry

On all three sides of a right triangle $ABC$ external squares are constructed; their centers denoted by $D$, $E$, $F$. Show that the ratio of the area of triangle $DEF$ to the area of triangle $ABC$ is: a) [i](2 points)[/i] greater than $1$; b) [i](2 points)[/i] at least $2$.