Olympiad problems

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

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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

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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

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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

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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

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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]

2010 F = Ma, 17

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Four masses $m$ are arranged at the vertices of a tetrahedron of side length $a$. What is the gravitational potential energy of this arrangement? (A) $-2\frac{Gm^2}{a}$ (B) $-3\frac{Gm^2}{a}$ (C) $-4\frac{Gm^2}{a}$ (D) $-6\frac{Gm^2}{a}$ (E) $-12\frac{Gm^2}{a}$

2015 IMO Shortlist, G5

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

TNO 2023 Senior, 4

Tags: combinatorics

In a country, there are $ n $ cities. Each pair of cities is connected either by a paved road or a dirt road. It is known that there exists a pair of cities such that it is impossible to travel between them using only paved roads. Show that, in this case, it is possible to travel between any two cities using only dirt roads.

2019 All-Russian Olympiad, 3

Tags: geometry , circumcircle

Circle $\Omega$ with center $O$ is the circumcircle of an acute triangle $\triangle ABC$ with $AB<BC$ and orthocenter $H$. On the line $BO$ there is point $D$ such that $O$ is between $B$ and $D$ and $\angle ADC= \angle ABC$ . The semi-line starting at $H$ and parallel to $BO$ wich intersects segment $AC$ , intersects $\Omega$ at $E$. Prove that $BH=DE$.