Olympiad problems

2021 All-Russian Olympiad, 5

Tags: algebra

The reals $b>0$ and $a$ are such that the quadratic $x^2+ax+b$ has two distinct real roots, exactly one of which lies in the interval $[-1;1]$. Prove that one of the roots lies in the interval $(-b;b)$.

2009 Peru IMO TST, 4

Tags: number theory

Show that there exist $2009$ consecutive positive integers such that for each of them the ratio between the largest and the smallest prime divisor is more than $20.$

2012 Romania National Olympiad, 2

Tags: matrix , vector , linear algebra

[color=darkred]Let $n$ and $k$ be two natural numbers such that $n\ge 2$ and $1\le k\le n-1$ . Prove that if the matrix $A\in\mathcal{M}_n(\mathbb{C})$ has exactly $k$ minors of order $n-1$ equal to $0$ , then $\det (A)\ne 0$ .[/color]

2000 Estonia National Olympiad, 1

Tags: algebra , percentage

The managing director of AS Mull, a brokerage company for soap bubbles, air castles and cheese holes, kissed the sales manager lazily, claiming that the company's sales volume in December had decreased by more than $10\%$ compared to October. Muugijuht, on the other hand, wrote in his quarterly report that although each, in the first half of the month, sales decreased compared to the second half of the previous month $30\%$ of the time, it increased in the second half of each month compared to the first half of the same month by $35\%$. Was the CEO wrong when the sales manager's report is true?

2012 Canada National Olympiad, 3

Tags: calculus , incenter , ellipse , angle bisector , geometry , conic

Let $ABCD$ be a convex quadrilateral and let $P$ be the point of intersection of $AC$ and $BD$. Suppose that $AC+AD=BC+BD$. Prove that the internal angle bisectors of $\angle ACB$, $\angle ADB$ and $\angle APB$ meet at a common point.

1997 All-Russian Olympiad Regional Round, 11.3

Tags: number theory , sum of digits

Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$

Russian TST 2021, P1

Tags: combinatorics , number theory

A machine accepts coins of $k{}$ values $1 = a_1 <\cdots < a_k$ and sells $k{}$ different drinks with prices $0<b_1 < \cdots < b_k$. It is known that if we start inserting coins into the machine in an arbitrary way, sooner or later the total value of the coins will be equal to the price of a drink. For which sets of numbers $(a_1,\ldots,a_k;b_1,\ldots,b_k)$ does this property hold?

2017 IFYM, Sozopol, 6

Tags: number theory

Find all triples $(x,y,z)$, $x,y,z\in \mathbb{Z}$ for which the number 2016 can be presented as $\frac{x^2+y^2+z^2}{xy+yz+zx}$.

2012 Oral Moscow Geometry Olympiad, 5

Tags: minimum , sum , distance , geometry , circles

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

2009 Spain Mathematical Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , octahedron , icosahedron , topology , combinatorics

Some edges are painted in red. We say that a coloring of this kind is [i]good[/i], if for each vertex of the polyhedron, there exists an edge which concurs in that vertex and is not painted red. Moreover, we say that a coloring where some of the edges of a regular polyhedron is [i]completely good[/i], if in addition to being [i]good[/i], no face of the polyhedron has all its edges painted red. What regular polyhedrons is equal the maximum number of edges that can be painted in a [i]good[/i] color and a [i]completely good[/i]? Explain your answer.

2023 LMT Fall, 3A

Tags: theme , geo

A rectangular tea bag $PART$ has a logo in its interior at the point $Y$ . The distances from $Y$ to $PT$ and $PA$ are $12$ and $9$ respectively, and triangles $\triangle PYT$ and $\triangle AYR$ have areas $84$ and $42$ respectively. Find the perimeter of pentagon $PARTY$. [i]Proposed by Muztaba Syed[/i] [hide=Solution] [i]Solution[/i]. $\boxed{78}$ Using the area and the height in $\triangle PYT$, we see that $PT = 14$, and thus $AR = 14$, meaning the height from $Y$ to $AR$ is $6$. This means $PA = TR = 18$. By the Pythagorean Theorem $PY=\sqrt{12^2+9^2} = 15$ and $YT =\sqrt{12^2 +5^2} = 13$. Combining all of these gives us an answer of $18+14+18+13+15 = \boxed{78}$. [/hide]

1973 Miklós Schweitzer, 10

Tags: trigonometry , probability and stats

Find the limit distribution of the sequence $ \eta_n$ of random variables with distribution \[ P \left( \eta_n\equal{}\arccos (\cos^2 \frac{(2j\minus{}1) \pi}{2n}) \right)\equal{}\frac 1n \;(j\equal{}1,2,...,n)\ .\] ($ \arccos(.)$ denotes the main value.) [i]B. Gyires[/i]

2003 National Olympiad First Round, 10

Tags: modular arithmetic , geometry , 3d geometry , algebra , factorization , sum of cubes

Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ \text{None of the preceding} $

1998 National Olympiad First Round, 1

Tags: trigonometry , quadratic

If $ \left|BC\right| \equal{} a$, $ \left|AC\right| \equal{} b$, $ \left|AB\right| \equal{} c$, $ 3\angle A \plus{} \angle B \equal{} 180{}^\circ$ and $ 3a \equal{} 2c$, then find $ b$ in terms of $ a$. $\textbf{(A)}\ \frac {3a}{2} \qquad\textbf{(B)}\ \frac {5a}{4} \qquad\textbf{(C)}\ a\sqrt {2} \qquad\textbf{(D)}\ a\sqrt {3} \qquad\textbf{(E)}\ \frac {2a\sqrt {3} }{3}$

2008 China Team Selection Test, 1

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , ratio , trigonometry

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

2006 India National Olympiad, 6

Tags: induction , inequalities

(a) Prove that if $n$ is a integer such that $n \geq 4011^2$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \] (b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \]

2023 JBMO Shortlist, N4

Tags: number theory

The triangle $ABC$ is sectioned by $AD,BE$ and $CF$ (where $D \in (BC), E \in (CA)$ and $F \in (AB)$) in seven disjoint polygons named [i]regions[/i]. In each one of the nine vertices of these regions we write a digit, such that each nonzero digit appears exactly once. We assign to each side of a region the lowest common multiple of the digits at its ends, and to each region the greatest common divisor of the numbers assigned to its sides. Find the largest possible value of the product of the numbers assigned to the regions.

2023 Regional Olympiad of Mexico Southeast, 4

Tags: algebra , fibonacci

Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$

PEN O Problems, 42

Tags:

Let $N_{n}$ denote the number of ordered $n$-tuples of positive integers $(a_{1},a_{2},\ldots,a_{n})$ such that \[1/a_{1}+1/a_{2}+\ldots+1/a_{n}=1.\] Determine whether $N_{10}$ is even or odd.

2011 Gheorghe Vranceanu, 4

Tags: number theory

Prove that for any natural number $ n $ there are $ n $ consecutive numbers, each one of these numbers having the following property: the sum of the positive divisors of a number $ x $ is greater than $ 2x. $

LMT Theme Rounds, 4

Tags:

A male volcano is in the shape of a hollow cone with the point side up, but with everything above a height of 6 meters removed. The resulting shape has a bottom radius of 10 meters and a top radius of 7 meters, with a height of 6 meters. He sat above his bay, watching all the couples play. His lava grew and grew until he was half full of lava. Then, he erupted, lowering the height of the lava to 2 meters. What fraction of the lava remained in the volcano? [i]Proposed by Matthew Weiss

2023 Dutch BxMO TST, 3

Tags: combinatorics

We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.

2008 Mongolia Team Selection Test, 3

Tags: circumcircle , power of a point , radical axis , geometry

Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$ varies on $ \omega$

2025 CMIMC Team, 7

Tags: team

The binomial coefficient $\tbinom{n}{k}$ can be defined as the coefficient of $x^k$ in the expansion of $(1+x)^n.$ Similarly, define the trinomial coefficient $\tbinom{n}{k}_3$ as the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n.$ Determine the number of integers $k$ with $0 \le k \le 4048$ such that $\tbinom{2024}{k}_3 \equiv 1 \pmod{3}.$

2005 Georgia Team Selection Test, 2

Tags: trigonometry , geometry , geometric transformation , reflection

In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.