Olympiad problems

2023 Middle European Mathematical Olympiad, 2

Tags: geometry , combinatorics

Find all positive integers $n \geq 3$, for which it is possible to draw $n$ chords on a circle, with their $2n$ endpoints being pairwise distinct, such that each chords intersects exactly $k$ others for: (a) $k=n-2$, (b) $k=n-3$.

2020 Sharygin Geometry Olympiad, 11

Tags: geometry

Let $ABC$ be a triangle with $\angle A=60^{\circ}$, $AD$ be its bisector, and $PDQ$ be a regular triangle with altitude $DA$. The lines $PB$ and $QC$ meet at point $K$. Prove that $AK$ is a symmedian of $ABC$.

1964 Bulgaria National Olympiad, Problem 2

Tags:

Find all $n$-tuples of reals $x_1,x_2,\ldots,x_n$ satisfying the system: $$\begin{cases}x_1x_2\cdots x_n=1\\x_1-x_2x_3\cdots x_n=1\\x_1x_2-x_3x_4\cdots x_n=1\\\vdots\\x_1x_2\cdots x_{n-1}-x_n=1\end{cases}$$

2021 Purple Comet Problems, 4

Tags:

A building contractor needs to pay his $108$ workers $\$200$ each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two $\$100$ bills. Find the number of workers who get paid with four $\$50$ bills.

1953 AMC 12/AHSME, 33

Tags: geometry , perimeter

The perimeter of an isosceles right triangle is $ 2p$. Its area is: $ \textbf{(A)}\ (2\plus{}\sqrt{2})p \qquad\textbf{(B)}\ (2\minus{}\sqrt{2})p \qquad\textbf{(C)}\ (3\minus{}2\sqrt{2})p^2\\ \textbf{(D)}\ (1\minus{}2\sqrt{2})p^2 \qquad\textbf{(E)}\ (3\plus{}2\sqrt{2})p^2$

2008 Estonia Team Selection Test, 1

Tags: combinatorics

There are $2008$ participants in a programming competition. In every round, all programmers are divided into two equal-sized teams. Find the minimal number of rounds after which there can be a situation in which every two programmers have been in different teams at least once.

2018-2019 SDML (High School), 1

Tags: factorial

Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.

2016 Korea - Final Round, 3

Tags: number theory , diophantine equation

Prove that for all rationals $x,y$, $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.

2006 Hong Kong TST., 6

Tags: induction

Find $2^{2006}$ positive integers satisfying the following conditions. (i) Each positive integer has $2^{2005}$ digits. (ii) Each positive integer only has 7 or 8 in its digits. (iii) Among any two chosen integers, at most half of their corresponding digits are the same.

2003 AMC 8, 14

Tags:

In this addition problem, each letter stands for a different digit. $ \setlength{\tabcolsep}{0.5mm}\begin{array}{cccc}&T & W & O\\ \plus{} &T & W & O\\ \hline F& O & U & R\end{array} $ If T = 7 and the letter O represents an even number, what is the only possible value for W? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

2008 Hong Kong TST, 3

Tags: number theory

Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$

2004 National Olympiad First Round, 18

Tags:

How many consequtive numbers are there in the set of positive integers in which powers of all prime factors in their prime factorizations are odd numbers? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 15 $

2011-2012 SDML (High School), 10

Tags:

Let $X=\left\{1,2,3,4,5,6\right\}$. How many non-empty subsets of $X$ do not contain two consecutive integers? $\text{(A) }16\qquad\text{(B) }18\qquad\text{(C) }20\qquad\text{(D) }21\qquad\text{(E) }24$

2015 Kyiv Math Festival, P4

Tags: geometry , perimeter , circumcenter

Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$

2012 Iran MO (3rd Round), 6

Tags: combinatorics , geometry

[b]a)[/b] Prove that $a>0$ exists such that for each natural number $n$, there exists a convex $n$-gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$. [b]b)[/b] Prove that there exists $b>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$. [b]c)[/b] Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$. [i]Proposed by Mostafa Eynollahzade[/i]

2003 Bulgaria Team Selection Test, 6

Tags: number theory

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$

1954 Moscow Mathematical Olympiad, 275

Tags: symmetry , heptagon , geometry

How many axes of symmetry can a heptagon have?

2016 Saudi Arabia GMO TST, 1

Tags: geometry , orthocenter , midpoint , parallel , circumcenter

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the following assertions: a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$. b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.

2001 AIME Problems, 7

Tags: geometry , incenter , ratio , inradius , number theory

Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2011 CIIM, Problem 6

Tags:

Let $\Gamma$ be the branch $x> 0$ of the hyperbola $x^2 - y^2 = 1.$ Let $P_0, P_1,..., P_n$ different points of $\Gamma$ with $P_0 = (1, 0)$ and $P_1 = (13/12, 5/12)$. Let $t_i$ be the tangent line to $\Gamma$ at $P_i$. Suppose that for all $i \geq 0$ the area of the region bounded by $t_i, t_{i +1}$ and $\Gamma$ is a constant independent of $i$. Find the coordinates of the points $P_i$.

2017 CCA Math Bonanza, L2.2

Tags:

Non-degenerate triangle $ABC$ has $AB=20$, $AC=17$, and $BC=n$, an integer. How many possible values of $n$ are there? [i]2017 CCA Math Bonanza Lightning Round #2.2[/i]

2015 Azerbaijan IMO TST, 2

Tags: algebra , function

Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$

2023 Polish Junior Math Olympiad Finals, 3.

Tags: geometry

Triangle $ABC$ is given, where $AC<BC$ and $\angle ACB=60^\circ\!\!.$ Point $D$, distinct from $A$, lies on the segment $AC$ such that $AB=BD$, and point $E$, distinct from $B$, lies on the line $BC$ such that $AB=AE$. Prove that $\angle DEC=30^\circ$.

1987 ITAMO, 4

Tags: equation , algebra , set

Given $I_0 = \{-1,1\}$, define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$, where $y$ ranges over all elements of $I_{n-1}$. Determine the union of the sets $I_n$ over all nonnegative integers $n$.

2017 Online Math Open Problems, 29

Tags:

Let $p = 2017$. If $A$ is an $n\times n$ matrix composed of residues $\pmod{p}$ such that $\det A\not\equiv 0\pmod{p}$ then let $\text{ord}(A)$ be the minimum integer $d > 0$ such that $A^d\equiv I\pmod{p}$, where $I$ is the $n\times n$ identity matrix. Let the maximum such order be $a_n$ for every positive integer $n$. Compute the sum of the digits when $\sum_{k = 1}^{p + 1} a_k$ is expressed in base $p$. [i]Proposed by Ashwin Sah[/i]