Problems

In a convex quadrilateral $ABCD$, all the triangles $\mathrm{△}ABC$, $\mathrm{△}BCD$, $\mathrm{△}CDA$ and $\mathrm{△}DAB$ have equal perimeters. Show that $ABCD$ is a rectangle.

Find the last two digits of the number $${33333333333333333347}^4-{11111111111111111147}^4$$

Replace all stars with ”+” or ”$\times$” signs so the equation holds: $$1\ast 2\ast 3\ast 4\ast 5\ast 6=100$$ Extra brackets may be added if necessary. Please write down the expression into the answer box.

In how many ways can one change $\text{pounds}2$ into coins worth $50$p, $20$p and $10$p? One does not necessarily need to use all available coin types, i.e. having $5$ coins of $20$p and $10$ coins of $10$p is allowed.

Consider two congruent triangles $ABC$ and $A_1{B}_1{C}_1$. We draw a point $M$ on the side $BC$ and a point $M_1$ on the side $B_1{C}_1$ such that the ratio of lengths $BM:MC$ is equal to the ratio of lengths $B_1{M}_1:M_1{C}_1$. Prove that $AM=A_1{M}_1$.

We call a median the segment from the vertex of a triangle to the midpoint of the opposite side. Prove that in two congruent triangles, the corresponding medians are of equal length.

We call a bisector the segment from the vertex of a triangle to the opposite side which divides in half the angle next to the starting vertex. Prove that in two congruent triangles, the corresponding bisectors are of equal length.

$7$ identical hexagons are arranged in a pattern on the picture below. If each hexagon has an area of $8$, what is the area of the triangle $\mathrm{△}ABC$?

In the triangle $ABC$ the bisector $BD$ coincides with the height. Prove that $AB=BC$.

In the triangle $ABC$ the median $BD$ coincides with the height. Prove that $AB=BC$.