Olympiad problems

1972 All Soviet Union Mathematical Olympiad, 160

Tags: combinatorial geometry

Given $50$ segments on the line. Prove that one of the following statements is valid: 1. Some $8$ segments have the common point. 2. Some $8$ segments do not intersect each other.

1971 Miklós Schweitzer, 7

Tags: combinatorics

Let $ n \geq 2$ be an integer, let $ S$ be a set of $ n$ elements, and let $ A_i , \; 1\leq i \leq m$, be distinct subsets of $ S$ of size at least $ 2$ such that \[ A_i \cap A_j \not\equal{} \emptyset, A_i \cap A_k \not\equal{} \emptyset, A_j \cap A_k \not\equal{} \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not\equal{} \emptyset \ .\] Show that $ m \leq 2^{n\minus{}1}\minus{}1$. [i]P. Erdos[/i]

2017 Taiwan TST Round 2, 2

Tags: inequalities

Let $a,b,c,d$ be positive real numbers satisfying $a+b+c+d=4$. Prove that $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\geq 4+(a-d)^2$$

2010 Danube Mathematical Olympiad, 1

Tags: geometry

Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.

1993 Poland - Second Round, 2

Tags: geometry , fixed , fixed point , concurrency

Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.

2017 ISI Entrance Examination, 4

Tags: geometry

Let $S$ be a square formed by the four vertices $(1,1),(1.-1),(-1,1)$ and $(-1,-1)$. Let the region $R$ be the set of points inside $S$ which are closer to the center than any of the four sides. Find the area of the region $R$.

2015 Online Math Open Problems, 14

Tags:

Let $ABCD$ be a square with side length $2015$. A disk with unit radius is packed neatly inside corner $A$ (i.e. tangent to both $\overline{AB}$ and $\overline{AD}$). Alice kicks the disk, which bounces off $\overline{CD}$, $\overline{BC}$, $\overline{AB}$, $\overline{DA}$, $\overline{DC}$ in that order, before landing neatly into corner $B$. What is the total distance the center of the disk travelled? [i]Proposed by Evan Chen[/i]

2004 Regional Olympiad - Republic of Srpska, 4

Tags: combinatorics

An $8\times8$ chessboard is completely tiled by $2\times1$ dominoes. Prove that there exist a king's tour of that chessboard such that every cell of the board is visited exactly once and such that king goes domino by domino, i.e. if king moves to the first cell of a domino, it must move to another cell in the next move. (King doesn't have to come back to the initial cell. King is an usual chess piece.)

2011 Akdeniz University MO, 5

Tags: geometry , circumcircle , perpendicular bisector

Let $ABC$ be an acute-angled triangle with $H$ orthocenter, $O$ circumcenter. $[AH]$'s perpendicular bisector intersects with $[AB]$ and $[AC]$ at $D$ and $E$ respectively. Prove that $$\angle ADE =\angle BDO$$

2017 Estonia Team Selection Test, 7

Tags: combinatorics , square table , table

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

The Golden Digits 2024, P2

Tags: geometry

Let $ABC$ be a triangle and $P$ a point in its interior. Circle $\Gamma_A$ is considered such that it is tangent to rays $(PB$ and $(PC$. Define similarly $\Gamma_B$ and $\Gamma_C$. Let $\ell_A\neq PA$ be the other common internal tangent of $\Gamma_B$ and $\Gamma_C$. Prove that $\ell_A$, $\ell_B$ and $\ell_C$ meet at a point. [i]Proposed by Andrei Vila[/i]

2007 Princeton University Math Competition, 3

Tags: geometry , analytic geometry , quadratic , calculus , integration , algebra , quadratic formula

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2006 Baltic Way, 8

Tags: induction , combinatorics

The director has found out that six conspiracies have been set up in his department, each of them involving exactly $3$ persons. Prove that the director can split the department in two laboratories so that none of the conspirative groups is entirely in the same laboratory.

1992 Baltic Way, 1

Tags: number theory

Let $p,q$ be two consecutive odd prime numbers. Prove that $p+q$ is a product of at least $3$ natural numbers greater than $1$ (not necessarily different).

2023 Durer Math Competition Finals, 2

Tags: number theory

When Andris entered the room, there were the numbers $3$ and $24$ on the board. In one step, if there are the (not necessarily different) numbers $k$ and $n$ on the board already, then Andris can write the number$ kn + k + n$ on the board, too. a) Can Andris write the number $9999999$ on the board after a few moves? b) What if he wants to get $99999999$? c) And what about $48999999$?

2017 AMC 10, 11

Tags: ratio

At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it? $\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$

1970 IMO Longlists, 12

Tags: algebra , polynomial , number theory

Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$. $(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$, where the $\pm$ signs are independent of each other. $(b)$ Generalize the result to every prime number.

2023 Singapore Senior Math Olympiad, 3

Tags: combinatorics

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

1999 AIME Problems, 9

Tags: function , complex number , number theory , relatively prime

A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2002 India IMO Training Camp, 19

Tags: geometry , circumcircle , trigonometry , triangle

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

1988 Putnam, A2

Tags:

A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$.

2021 AMC 10 Spring, 9

Tags: sfft

What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$

2015 Peru Cono Sur TST, P5

Tags: number theory

Find the smallest term of the sequence $a_1, a_2, a_3, \ldots$ defined by $a_1=2014^{2015^{2016}}$ and $$ a_{n+1}= \begin{cases} \frac{a_n}{2} & \text{ if } a_n \text{ is even} \\ a_n + 7 & \text{ if } a_n \text{ is odd} \\ \end{cases} $$

2006 Estonia Math Open Senior Contests, 8

Tags: geometry

Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.

2011 Northern Summer Camp Of Mathematics, 2

Tags: function , number theory

Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and \[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]