Olympiad problems

2010 Contests, 1

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

2016 India IMO Training Camp, 2

Tags: function , functional equation , algebra

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.

2003 Olympic Revenge, 5

Tags: number theory , prime numbers , prime , Divisibility

Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

2012 Today's Calculation Of Integral, 821

Tags: calculus , integration , logarithms , inequalities , calculus computations

Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$

2013 Moldova Team Selection Test, 3

Tags: geometry , parallelogram , reflection , IMO Shortlist , moving points , ptolemy sinus lemma , Hi

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

2024 Indonesia MO, 6

Tags: geometry , polygon , Inequality , convex , inequalities , Indonesia , Indonesia MO

Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides). For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that \[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.

2022 Junior Macedonian Mathematical Olympiad, P4

Tags: combinatorics

An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown on the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for $k \in \left \{1,2,3,4,5 \right \}$. The covering satisfies the following conditions: $1)$ It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid. $2)$ Every unit triangle in the grid is covered, no two figures overlap and every figure is fully contained in $T$. Determine the smallest possible number of figures of type $1$ that can be used in such a covering. [i]Proposed by Ilija Jovcheski[/i]

2008 Stanford Mathematics Tournament, 13

Tags:

Let N be the number of distinct rearrangements of the 34 letters in SUPERCALIFRAGILISTICEXPIALIDOCIOUS. How many positive factors does N have?

1960 Putnam, B4

Tags: Putnam , arithmetic sequence , powers

Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.

2011 Kazakhstan National Olympiad, 1

Tags: ratio , trigonometry , geometry unsolved , geometry

Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.

1987 Mexico National Olympiad, 2

Tags: number theory , number of divisors

How many positive divisors does number $20!$ have?

1998 Bulgaria National Olympiad, 1

Tags: combinatorics unsolved , combinatorics

Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.

2018 Hanoi Open Mathematics Competitions, 10

Tags: Coloring , combinatorics

The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level (Figure). We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: (a) The top level is painted with only one color. (b) The $3$ doorways at the front on the second level are painted with the same color. (c) The $3$ doorways at the front on the first level are painted with the same color. (d) Each of the remaining $14$ doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors. How many ways are there to paint the first level? How many ways are there to paint the entire tower model? [img]https://cdn.artofproblemsolving.com/attachments/f/9/2249f8595a8efe711680f3dfb8ff959c140a21.png[/img]

2023 Princeton University Math Competition, B2

Tags: algebra

The sum $$\sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1}$$ can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find the remainder when $a+b$ is divided by $1000.$

1993 Korea - Final Round, 6

Tags: ratio , geometry , geometry proposed

Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and ﬁnd the ratio of their areas.

2001 Stanford Mathematics Tournament, 7

Tags: Stanford , college , geometry , geometric transformation , reflection

The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.

2017 Puerto Rico Team Selection Test, 1

Tags: Sum , algebra , functional equation , functional

Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.

1927 Eotvos Mathematical Competition, 2

Tags: number theory , Digits

Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.

2011 Today's Calculation Of Integral, 683

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$. [i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]

1997 Akdeniz University MO, 5

Tags: geometric inequality , geometry

A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that $$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$

1998 Harvard-MIT Mathematics Tournament, 1

Tags: geometry

Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!) can be inscribed in a circle; with $\angle LAX = 20^{\circ}$ and $\angle AXE = 100^{\circ}:$

1999 Junior Balkan MO, 1

Tags: algorithm

Let $ a,b,c,x,y$ be five real numbers such that $ a^3 \plus{} ax \plus{} y \equal{} 0$, $ b^3 \plus{} bx \plus{} y \equal{} 0$ and $ c^3 \plus{} cx \plus{} y \equal{} 0$. If $ a,b,c$ are all distinct numbers prove that their sum is zero. [i]Ciprus[/i]

2016 May Olympiad, 1

Tags: number theory , Digits

We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.

2004 China Team Selection Test, 1

Tags: function , quadratics , algebra unsolved , algebra

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

Cono Sur Shortlist - geometry, 2003.G2

Tags: tangent circles , diameter , geometry

The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$ the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$. Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.