Olympiad problems

2008 Sharygin Geometry Olympiad, 2

(V.Protasov, 8) For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?

2008 Hong Kong TST, 1

Tags: inequalities , combinatorics unsolved , combinatorics

In a school there are $ 2008$ students. Students are members of certain committees. A committee has at most $ 1004$ members and every two students join a common committee. (i) Determine the smallest possible number of committees in the school. (ii) If it is further required that the union of any two committees consists of at most $ 1800$ students, will your answer in (i) still hold?

2024 Indonesia TST, C

Tags: combinatorics , maximization , Indonesia , Indonesia TST

Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.

2007 Germany Team Selection Test, 3

Tags: ratio , geometry , circumcircle , geometry unsolved

In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?

2019 SG Originals, Q4

Tags: number theory , prime numbers , algebra , polynomial

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

2017-2018 SDML (Middle School), 15

Tags: SDML Middle School , 2017-18 SDML Round 1

For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$. For how many positive integers $x \leq 100$ is the value of $f(x)$ odd? $\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$

2009 Today's Calculation Of Integral, 405

Tags: calculus , integration , trigonometry , calculus computations

Calculate $ \displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|$.

2023 CIIM, 3

Tags: linear algebra , matrix , Sequences , symmetric matrix , eigenvalue and eigenvector

Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ such that $A_{n+1} = f(A_n)$ for $n \geq 0$. If the matrix $A_0$ has no zero entries, determine the maximum number of indices $j \geq 0$ for which the matrix $A_j$ has any null entries.

2019 AMC 10, 15

Tags: AMC , AMC 10 , AMC 10 B

Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of $T_1$ and $T_2$? $\textbf{(A) }\frac{28}3\qquad\textbf{(B) }10\qquad\textbf{(C) }\frac{32}3\qquad\textbf{(D) }\frac{34}3\qquad\textbf{(E) }12$

2022/2023 Tournament of Towns, P2

Tags: geometry , sidelengths

Medians $BK{}$ and $CN{}$ of triangle $ABC$ intersect at $M{}.$ Consider quadrilateral $ANMK$ and find the maximum possible number of its sides having length 1. [i]Egor Bakaev[/i]

the 5th XMO, 1

Tags: geometry , Concyclic

Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$. a) Prove that $\vartriangle PQH \sim \vartriangle AKV$. b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide] [hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img] [url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]

2019 AIME Problems, 3

Tags: AMC , AIME , AIME I , geometry , 2019 AIME I , 2019 AMC , area

In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.

2008 Flanders Math Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , pyramid

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

2009 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , quartet

For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$

1953 AMC 12/AHSME, 49

Tags: analytic geometry , geometry , geometric transformation , reflection

The coordinates of $ A,B$ and $ C$ are $ (5,5),(2,1)$ and $ (0,k)$ respectively. The value of $ k$ that makes $ \overline{AC}\plus{}\overline{BC}$ as small as possible is: $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac{1}{2} \qquad\textbf{(C)}\ 3\frac{6}{7} \qquad\textbf{(D)}\ 4\frac{5}{6} \qquad\textbf{(E)}\ 2\frac{1}{7}$

1994 Nordic, 2

Tags: lattice points , combinatorics

We call a finite plane set $S$ consisting of points with integer coefficients a two-neighbour set, if for each point $(p, q)$ of $S$ exactly two of the points $(p +1, q), (p, q +1), (p-1, q), (p, q-1)$ belong to $S$. For which integers $n$ there exists a two-neighbour set which contains exactly $n$ points?

ICMC 8, 2

Tags: college contests

Alice and the Mad Hatter are playing a game. At the start of the game, three $2024$’s are written on the blackboard. Then, Alice and the Mad Hatter alternate turns, with the Mad Hatter starting. On the Mad Hatter’s turn, he must pick one of the numbers on the blackboard and increase it by $1$. On Alice’s turn, she must: - pick one of the numbers on the blackboard and decrease it by 1, and then - replace the two numbers $a$ and $b$ on the blackboard which were not chosen by the Mad Hatter on the previous turn with $\sqrt{ab}$. Alice wins if, on the start of her turn, any of the three numbers are less than $1$. Can the Mad Hatter prevent Alice from winning?

2022 LMT Fall, 10

Tags: algebra

Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $. $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

1997 Tournament Of Towns, (548) 2

Tags: number theory , Diophantine equation , diophantine

Prove that the equation $x^2 + y^2 - z^2 = 1997$ has infinitely many solutions in integers $x$, $y$ and $z$. (N Vassiliev)

2022 DIME, 2

Tags: DIME P2

Let $P(x) = x^2-1$ be a polynomial, and let $a$ be a positive real number satisfying$$P(P(P(a))) = 99.$$ The value of $a^2$ can be written as $m+\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. [i]Proposed by [b]HrishiP[/b][/i]

2009 Bundeswettbewerb Mathematik, 4

Tags: combinatorial geometry , combinatorics , diagonals

How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?

2005 Slovenia National Olympiad, Problem 3

Tags: geometry , Triangle

In an isosceles triangle $ABC$ with $AB = AC$, $D$ is the midpoint of $AC$ and $E$ is the projection of $D$ onto $BC$. Let $F$ be the midpoint of $DE$. Prove that the lines $BF$ and $AE$ are perpendicular if and only if the triangle $ABC$ is equilateral.

2025 Bangladesh Mathematical Olympiad, P10

Tags: geometry

In $\triangle ABC$, $DB$ and $DC$ are tangent to the circumcircle of $\triangle ABC$. Let $B'$ be the reflection of $B$ with respect to the line $AC$ and similarly define $C'$. If line $BC$ intersects the circumcircle of $\triangle DB'C'$ at $E$ and $F$, prove that $AE = AF$.

2004 Bosnia and Herzegovina Team Selection Test, 2

Tags: number theory , geometry , area

Determine whether does exists a triangle with area $2004$ with his sides positive integers.

2008 AIME Problems, 10

Tags: analytic geometry , AMC

The diagram below shows a $ 4\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors. [asy]unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j));[/asy]Define a [i]growing path[/i] to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $ m$ be the maximum possible number of points in a growing path, and let $ r$ be the number of growing paths consisting of exactly $ m$ points. Find $ mr$.