Olympiad problems

2012 AMC 10, 23

Tags: geometry , 3d geometry , tetrahedron , pyramid , hmmt , analytic geometry , circumcircle

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object? $ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $

1997 Korea National Olympiad, 8

Tags: arithmetic sequence

For any positive integers $x,y,z$ and $w,$ prove that $x^2,y^2,z^2$ and $w^2$ cannot be four consecutive terms of arithmetic sequence.

2013 Kyiv Mathematical Festival, 1

Tags: combinatorics

There are $24$ apples in $4$ boxes. An optimistic worm is convinced that he can eat no more than half of the apples such that there will be $3$ boxes with equal number of apples. Is it possible that he is wrong?

2006 VJIMC, Problem 2

Tags: group theory , abstract algebra

Let $(G,\cdot)$ be a finite group of order $n$. Show that each element of $G$ is a square if and only if $n$ is odd.

2010 IMO, 1

Tags: algebra , functional equation , fe

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

2015 AMC 12/AHSME, 7

Tags: symmetry

A regular $15$-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$? $\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }32\qquad\textbf{(D) }39\qquad\textbf{(E) }54$

2016 Nordic, 3

Tags: functional equation , function

Find all $a\in\mathbb R$ for which there exists a function $f\colon\mathbb R\rightarrow\mathbb R$, such that (i) $f(f(x))=f(x)+x$, for all $x\in\mathbb R$, (ii) $f(f(x)-x)=f(x)+ax$, for all $x\in\mathbb R$.

2020 Taiwan TST Round 2, 2

Tags: geometry

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

1990 Polish MO Finals, 1

Tags: geometry

A triangle whose all sides have length not smaller than $1$ is inscribed in a square of side length $1$. Prove that the center of the square lies inside the triangle or on its boundary.

1985 Brazil National Olympiad, 3

Tags: inequalities , geometry , perimeter , cyclic quadrilateral

A convex quadrilateral is inscribed in a circle of radius $1$. Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$.

1998 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory , power of 2

Let $M$ be a subset of $\{1,2,..., 1998\}$ with $1000$ elements. Prove that it is always possible to find two elements $a$ and $b$ in $M$, not necessarily distinct, such that $a + b$ is a power of $2$.

2024 JHMT HS, 13

Tags: number theory , quadratic

Compute the largest nonnegative integer $T \leq 30$ that is the remainder when $T^2 + 4$ is divided by $31$.

Kyiv City MO Seniors 2003+ geometry, 2017.10.3

Tags: geometry , square , equal segments

Given the square $ABCD$. Let point $M$ be the midpoint of the side $BC$, and $H$ be the foot of the perpendicular from vertex $C$ on the segment $DM$. Prove that $AB = AH$. (Danilo Hilko)

2003 All-Russian Olympiad, 1

Tags: algebra

Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b \in M$, the number $a^2 +b\sqrt 2$ is rational. Prove that $a\sqrt 2$ is rational for all $a \in M.$

2006 MOP Homework, 1

Tags: inequalities , algebra

Let a,b, and c be positive reals. Prove: $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

2005 USAMTS Problems, 4

Tags: algebra , polynomial

Find, with proof, all triples of real numbers $(a, b, c)$ such that all four roots of the polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+b$ are positive integers. (The four roots need not be distinct.)

2023 Puerto Rico Team Selection Test, 4

Tags: number theory

Find all positive integers $n$ such that: $$n = a^2 + b^2 + c^2 + d^2,$$ where $a < b < c < d$ are the smallest divisors of $n$.

2001 Taiwan National Olympiad, 1

Tags: algebra , polynomial , number theory

Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.

2007 IMS, 4

Tags: linear algebra

Prove that: \[\det(A)=\frac{1}{n!}\left| \begin{array}{llllll}\mbox{tr}(A) & 1 & 0 & \ldots & \ldots & 0 \\ \mbox{tr}(A^{2}) & \mbox{tr}(A) & 2 & 0 & \ldots & 0 \\ \mbox{tr}(A^{3}) & \mbox{tr}(A^{2}) & \mbox{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \mbox{tr}(A^{n}) & \mbox{tr}(A^{n-1}) & \mbox{tr}(A^{n-2}) & \ldots & \ldots & \mbox{tr}(A) \end{array}\right|\]

2013 HMNT, 5

Tags: geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ and $E$ be the midpoints of segments $AB$ and $AC$, respectively. Suppose that there exists a point$ F$ on ray $\overrightarrow{DE}$ outside of $ABC$ such that triangle $BFA$ is similar to triangle $ABC$. Compute $\frac{AB}{BC}$

2023 New Zealand MO, 7

Tags: inequalities , combinatorics , number theory

Let $n,m$ be positive integers. Let $A_1,A_2,A_3, ... ,A_m$ be sets such that $A_i \subseteq \{1, 2, 3, . . . , n\}$ and $|A_i| = 3$ for all $i$ (i.e. $A_i$ consists of three different positive integers each at most $n$). Suppose for all $i < j$ we have $|A_i \cap A_j | \le 1$ (i.e. $A_i$ and $A_j$ have at most one element in common). (a) Prove that $m \le \frac{n(n-1)}{ 6}$ . (b) Show that for all $n \ge3$ it is possible to have $m \ge \frac{(n-1)(n-2)}{ 6}$ .

2018 PUMaC Individual Finals B, 3

Tags: geometry

Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.

1996 AMC 8, 9

Tags:

If $5$ times a number is $2$, then $100$ times the reciprocal of the number is $\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500$

1988 IMO Longlists, 52

Tags: geometric transformation , reflection , geometry

$ ABCD$ is a quadrilateral. $ A'BCD'$ is the reflection of $ ABCD$ in $ BC,$ $ A''B'CD'$ is the reflection of $ A'BCD'$ in $ CD'$ and $ A''B''C'D'$ is the reflection of $ A''B'CD'$ in $ D'A''.$ Show that; if the lines $ AA''$ and $ BB''$ are parallel, then ABCD is a cyclic quadrilateral.

2022 Francophone Mathematical Olympiad, 3

Tags: geometry

Let $\triangle ABC$ a triangle, and $D$ the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. the line parallel to $AC$ passing by the point $B$, intersect the line $AD$ at $X$. the line parallel to $CX$ passing by the point $B$, intersect $AC$ at $Y$. $E = (AYB) \cap BX$ . prove that $C$ , $D$ and $E$ collinear.